{"id":8361,"date":"2026-05-19T09:59:14","date_gmt":"2026-05-19T09:59:14","guid":{"rendered":"https:\/\/mrbartonmaths.com\/?page_id=8361"},"modified":"2026-05-19T18:00:21","modified_gmt":"2026-05-19T18:00:21","slug":"circle-theorems-interactive-tool","status":"publish","type":"page","link":"https:\/\/mrbartonmaths.com\/circle-theorems-interactive-tool\/","title":{"rendered":"Circle theorems interactive tool"},"content":{"rendered":"\n<style>\n#circth_k3Vn_app { font-family: Arial, Helvetica, sans-serif; max-width: 980px; margin: 24px auto; user-select: none; -webkit-user-select: none; position: relative; color: #2c3e50; line-height: 1.5; }\n#circth_k3Vn_app *, #circth_k3Vn_app *::before, #circth_k3Vn_app *::after { box-sizing: border-box; margin: 0; padding: 0; }\n\n#circth_k3Vn_app .circth_k3Vn_toolbar { display: flex; align-items: center; padding: 14px 8px 10px; }\n#circth_k3Vn_app .circth_k3Vn_title { font-size: 1.35em; font-weight: 800; color: #2c3e50; white-space: nowrap; }\n#circth_k3Vn_app .circth_k3Vn_title span { color: #718096; font-weight: 500; font-size: 0.7em; margin-left: 6px; }\n\n#circth_k3Vn_app .circth_k3Vn_pill { display: inline-flex; align-items: center; justify-content: center; gap: 6px; padding: 8px 16px; border: none; border-radius: 20px; font-size: 0.86em; font-weight: 700; font-family: inherit; cursor: pointer; transition: all 0.18s ease; white-space: nowrap; line-height: 1.3; }\n#circth_k3Vn_app .circth_k3Vn_pill_action { background: #edf2f7; color: #4a5568; }\n#circth_k3Vn_app .circth_k3Vn_pill_action:hover { background: #e2e8f0; color: #2d3748; }\n#circth_k3Vn_app .circth_k3Vn_pill_action:active { transform: scale(0.94); }\n#circth_k3Vn_app .circth_k3Vn_pill_action:disabled { opacity: 0.35; cursor: default; }\n#circth_k3Vn_app .circth_k3Vn_pill_toggle { background: #edf2f7; color: #718096; }\n#circth_k3Vn_app .circth_k3Vn_pill_toggle:hover { background: #e2e8f0; color: #4a5568; }\n#circth_k3Vn_app .circth_k3Vn_pill_toggle.circth_k3Vn_on { background: #2b6cb0; color: #fff; box-shadow: 0 2px 8px rgba(43,108,176,0.3); }\n#circth_k3Vn_app .circth_k3Vn_pill_toggle.circth_k3Vn_on:hover { background: #2c5282; }\n#circth_k3Vn_app .circth_k3Vn_pill_primary { background: #2c3e50; color: #fff; padding: 10px 22px; font-size: 0.92em; }\n#circth_k3Vn_app .circth_k3Vn_pill_primary:hover { background: #1a202c; }\n#circth_k3Vn_app .circth_k3Vn_pill_primary.circth_k3Vn_on { background: #c05621; }\n#circth_k3Vn_app .circth_k3Vn_pill_primary.circth_k3Vn_on:hover { background: #9c4221; }\n#circth_k3Vn_app .circth_k3Vn_pill_sep { width: 1px; height: 22px; background: #dce1e6; flex-shrink: 0; }\n\n\/* Secondary toggle \u2014 smaller, lighter for less-important options *\/\n#circth_k3Vn_app .circth_k3Vn_pill_secondary { padding: 6px 12px; font-size: 0.78em; background: transparent; color: #a0aec0; border: 1px solid #e2e8f0; }\n#circth_k3Vn_app .circth_k3Vn_pill_secondary:hover { background: #f7fafc; 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14px; background: #f7fafc; border: 2px solid #e2e8f0; border-radius: 12px; margin-bottom: 12px; flex-wrap: wrap; }\n#circth_k3Vn_app .circth_k3Vn_control_primary { display: flex; align-items: center; gap: 8px; }\n#circth_k3Vn_app .circth_k3Vn_control_spacer { flex: 1; min-width: 8px; }\n#circth_k3Vn_app .circth_k3Vn_control_utilities { display: flex; align-items: center; gap: 6px; }\n\n#circth_k3Vn_app .circth_k3Vn_boards_row { display: flex; gap: 10px; }\n#circth_k3Vn_app .circth_k3Vn_board_column { display: flex; flex-direction: column; gap: 6px; flex: 1; min-width: 0; }\n#circth_k3Vn_app .circth_k3Vn_board_wrap { position: relative; background: #fafbfc; border: 2px solid #e2e8f0; border-radius: 12px; overflow: hidden; display: flex; align-items: center; justify-content: center; aspect-ratio: 1 \/ 1; max-height: 580px; margin: 0 auto; width: 100%; transition: border-color 0.2s, box-shadow 0.2s; }\n#circth_k3Vn_app .circth_k3Vn_boards_row.circth_k3Vn_split .circth_k3Vn_board_wrap { 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}\n\n\/* Tap hint below diagram *\/\n#circth_k3Vn_app .circth_k3Vn_tap_hint { text-align: center; font-size: 0.82em; color: #718096; padding: 8px 12px 2px; font-style: italic; }\n#circth_k3Vn_app .circth_k3Vn_tap_hint strong { color: #2c5282; font-weight: 700; font-style: normal; }\n\n.circth_k3Vn_circle_main { fill: rgba(43,108,176,0.04); stroke: #2c3e50; stroke-width: 2.5; }\n\/* Theorem 3: faint shading over the segment containing P *\/\n.circth_k3Vn_segment_fill { fill: rgba(192,86,33,0.08); stroke: none; pointer-events: none; transition: fill 0.2s ease; }\n\/* Theorem 4: faint quadrilateral interior fill *\/\n.circth_k3Vn_quad_fill { fill: rgba(43,108,176,0.06); stroke: none; pointer-events: none; }\n.circth_k3Vn_chord { stroke-width: 3; stroke-linecap: round; fill: none; transition: stroke-width 0.18s ease, filter 0.18s ease; }\n.circth_k3Vn_chord_centre { stroke: #2b6cb0; }\n.circth_k3Vn_chord_circ { stroke: #c05621; }\n.circth_k3Vn_chord_diameter { stroke: #2b6cb0; }\n.circth_k3Vn_chord_diameter_emph { stroke-width: 4; }\n\/* Same-segment theorem: AB chord is the reference, drawn lighter *\/\n.circth_k3Vn_chord_segment { stroke: #4a5568; stroke-dasharray: 6 4; stroke-width: 2.5; }\n\/* Q's arms \u2014 teal to distinguish from P's orange *\/\n.circth_k3Vn_chord_q { stroke: #2c7a7b; }\n\/* Cyclic quadrilateral sides \u2014 neutral so they don't compete with the angle colours *\/\n.circth_k3Vn_chord_quad { stroke: #2c3e50; }\n\/* Tangent-radius theorem: the tangent is purple to distinguish from any chord *\/\n.circth_k3Vn_chord_tangent { stroke: #6b46c1; stroke-width: 3; }\n\/* Alternate-segment theorem: the inscribed-angle arms PT and PA are green to match the angle pair *\/\n.circth_k3Vn_chord_alt { stroke: #2f855a; }\n\/* Highlighted state for chords when their associated angle arc is hovered *\/\n.circth_k3Vn_chord_highlighted { stroke-width: 5; filter: drop-shadow(0 0 6px currentColor); }\n.circth_k3Vn_centre_dot { fill: #2c3e50; stroke: #fff; stroke-width: 2; pointer-events: none; }\n.circth_k3Vn_drag_point { cursor: grab; transition: r 0.12s ease; }\n.circth_k3Vn_drag_point:hover { r: 13; }\n.circth_k3Vn_drag_point.circth_k3Vn_dragging { cursor: grabbing; }\n.circth_k3Vn_drag_glow { pointer-events: none; opacity: 0; transition: opacity 0.15s ease; }\n.circth_k3Vn_drag_glow.circth_k3Vn_active { opacity: 1; }\n.circth_k3Vn_point_label { font-family: Arial, Helvetica, sans-serif; font-size: 18px; font-weight: 800; pointer-events: none; }\n\n.circth_k3Vn_angle_arc { stroke-width: 2.5; cursor: pointer; transition: fill-opacity 0.15s, stroke-opacity 0.15s, stroke-width 0.15s; }\n.circth_k3Vn_angle_arc_centre { stroke: #2b6cb0; fill: rgba(43,108,176,0.18); }\n.circth_k3Vn_angle_arc_centre:hover { fill: rgba(43,108,176,0.32); stroke-width: 3.5; }\n.circth_k3Vn_angle_arc_circ { stroke: #c05621; fill: rgba(192,86,33,0.18); }\n.circth_k3Vn_angle_arc_circ:hover { fill: rgba(192,86,33,0.32); stroke-width: 3.5; }\n.circth_k3Vn_angle_arc_q { stroke: #2c7a7b; fill: rgba(44,122,123,0.18); }\n.circth_k3Vn_angle_arc_q:hover { fill: rgba(44,122,123,0.32); stroke-width: 3.5; }\n\/* Tangent-radius theorem: right angle is neutral dark (visual is the square shape, not colour) *\/\n.circth_k3Vn_angle_arc_right { stroke: #2c3e50; fill: rgba(44,62,80,0.12); }\n.circth_k3Vn_angle_arc_right:hover { fill: rgba(44,62,80,0.22); stroke-width: 3.5; }\n\/* Alternate-segment theorem: matching pair in green *\/\n.circth_k3Vn_angle_arc_alt { stroke: #2f855a; fill: rgba(47,133,90,0.18); }\n.circth_k3Vn_angle_arc_alt:hover { fill: rgba(47,133,90,0.32); stroke-width: 3.5; }\n\/* Two-tangents kite: angles at O and E (sum to 180\u00b0) in green *\/\n.circth_k3Vn_angle_arc_kite { stroke: #2f855a; fill: rgba(47,133,90,0.18); }\n.circth_k3Vn_angle_arc_kite:hover { fill: rgba(47,133,90,0.32); stroke-width: 3.5; }\n.circth_k3Vn_angle_arc.circth_k3Vn_revealed { stroke-width: 3; }\n.circth_k3Vn_angle_label { font-family: Arial, Helvetica, sans-serif; font-size: 18px; font-weight: 800; pointer-events: none; }\n.circth_k3Vn_angle_label_centre { fill: #2b6cb0; }\n.circth_k3Vn_angle_label_circ { fill: #c05621; }\n.circth_k3Vn_angle_label_q { fill: #2c7a7b; }\n.circth_k3Vn_angle_label_right { fill: #2c3e50; }\n.circth_k3Vn_angle_label_alt { fill: #2f855a; }\n.circth_k3Vn_angle_label_kite { fill: #2f855a; }\n\/* Two tangents theorem: tangent length labels in purple *\/\n.circth_k3Vn_angle_label_tangent { fill: #6b46c1; }\n.circth_k3Vn_angle_label_bg { fill: rgba(255,255,255,0.95); stroke-width: 1.5; pointer-events: none; }\n\n#circth_k3Vn_app .circth_k3Vn_theorem_panel { margin-top: 14px; border: 2px solid #e2e8f0; border-radius: 12px; overflow: hidden; background: #fff; }\n#circth_k3Vn_app .circth_k3Vn_theorem_toggle { width: 100%; display: flex; align-items: center; justify-content: space-between; padding: 14px 18px; background: #f7fafc; border: none; cursor: pointer; font-family: inherit; text-align: left; transition: background 0.15s ease; }\n#circth_k3Vn_app .circth_k3Vn_theorem_toggle:hover { background: #edf2f7; }\n#circth_k3Vn_app .circth_k3Vn_theorem_toggle_left { display: flex; align-items: center; gap: 12px; }\n#circth_k3Vn_app .circth_k3Vn_theorem_toggle_icon { width: 36px; height: 36px; border-radius: 50%; background: linear-gradient(135deg, #f6e05e 0%, #ecc94b 100%); color: #744210; display: flex; align-items: center; justify-content: center; font-size: 1.15em; flex-shrink: 0; box-shadow: 0 2px 6px rgba(236,201,75,0.4); }\n#circth_k3Vn_app .circth_k3Vn_theorem_toggle_text { font-size: 0.95em; font-weight: 700; color: #2c3e50; }\n#circth_k3Vn_app .circth_k3Vn_theorem_chevron { font-size: 1em; color: #a0aec0; transition: transform 0.25s ease; flex-shrink: 0; }\n#circth_k3Vn_app .circth_k3Vn_theorem_panel.circth_k3Vn_theorem_open .circth_k3Vn_theorem_chevron { transform: rotate(180deg); }\n#circth_k3Vn_app .circth_k3Vn_theorem_body { max-height: 0; overflow: hidden; transition: max-height 0.35s ease; background: linear-gradient(135deg, #fffbeb 0%, #fef5d3 100%); }\n#circth_k3Vn_app .circth_k3Vn_theorem_panel.circth_k3Vn_theorem_open .circth_k3Vn_theorem_body { max-height: 400px; }\n#circth_k3Vn_app .circth_k3Vn_theorem_body_inner { padding: 16px 20px; }\n#circth_k3Vn_app .circth_k3Vn_theorem_title { font-size: 1.05em; font-weight: 800; color: #744210; margin-bottom: 6px; }\n#circth_k3Vn_app .circth_k3Vn_theorem_text { font-size: 0.92em; color: #2c3e50; line-height: 1.55; }\n#circth_k3Vn_app .circth_k3Vn_theorem_text strong { color: #b7791f; }\n#circth_k3Vn_app .circth_k3Vn_theorem_invite { font-size: 0.88em; color: #744210; font-style: italic; margin-top: 8px; padding-top: 8px; border-top: 1px solid rgba(180,134,32,0.2); }\n\n\/* Investigations panel \u2014 exploration prompts for teachers and independent learners.\n   Visually distinct from the theorem panel (yellow = \"the answer\") by using a\n   soft teal\/cyan palette signalling exploration \/ inquiry. *\/\n#circth_k3Vn_app .circth_k3Vn_invest_panel { margin-top: 10px; border: 2px solid #e2e8f0; border-radius: 12px; overflow: hidden; background: #fff; }\n#circth_k3Vn_app .circth_k3Vn_invest_toggle { width: 100%; display: flex; align-items: center; justify-content: space-between; padding: 14px 18px; background: #f7fafc; border: none; cursor: pointer; font-family: inherit; text-align: left; transition: background 0.15s ease; }\n#circth_k3Vn_app .circth_k3Vn_invest_toggle:hover { background: #edf2f7; }\n#circth_k3Vn_app .circth_k3Vn_invest_toggle_left { display: flex; align-items: center; gap: 12px; }\n#circth_k3Vn_app .circth_k3Vn_invest_toggle_icon { width: 36px; height: 36px; border-radius: 50%; background: linear-gradient(135deg, #4fd1c5 0%, #38b2ac 100%); color: #fff; display: flex; align-items: center; justify-content: center; font-size: 1.05em; flex-shrink: 0; box-shadow: 0 2px 6px rgba(56,178,172,0.35); }\n#circth_k3Vn_app .circth_k3Vn_invest_toggle_text { font-size: 0.95em; 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rgba(56,178,172,0.4), transparent); margin: 16px 0; }\n\n@media (max-width: 640px) {\n    #circth_k3Vn_app .circth_k3Vn_title { font-size: 1em; }\n    #circth_k3Vn_app .circth_k3Vn_pill { padding: 6px 12px; font-size: 0.76em; }\n    #circth_k3Vn_app .circth_k3Vn_pill_primary { padding: 8px 16px; font-size: 0.82em; }\n    #circth_k3Vn_app .circth_k3Vn_tab { padding: 7px 12px; font-size: 0.78em; gap: 6px; }\n    #circth_k3Vn_app .circth_k3Vn_tab_bar { gap: 6px; padding: 4px 6px 10px; }\n    #circth_k3Vn_app .circth_k3Vn_tab_icon { width: 18px; height: 18px; }\n    #circth_k3Vn_app .circth_k3Vn_board_wrap { max-height: 380px; }\n    #circth_k3Vn_app .circth_k3Vn_boards_row.circth_k3Vn_split { flex-direction: column; }\n    #circth_k3Vn_app .circth_k3Vn_boards_row.circth_k3Vn_split .circth_k3Vn_board_wrap { max-height: 300px; }\n    #circth_k3Vn_app .circth_k3Vn_mini_strip { padding: 4px 6px; gap: 2px; }\n    #circth_k3Vn_app .circth_k3Vn_mini_icon { width: 26px; height: 26px; padding: 3px; }\n    #circth_k3Vn_app .circth_k3Vn_control_bar { padding: 8px; gap: 6px; }\n    #circth_k3Vn_app .circth_k3Vn_control_spacer { flex-basis: 100%; min-width: 0; height: 0; }\n    #circth_k3Vn_app .circth_k3Vn_theorem_toggle { padding: 12px 14px; }\n    #circth_k3Vn_app .circth_k3Vn_theorem_body_inner { padding: 12px 14px; }\n    #circth_k3Vn_app .circth_k3Vn_tap_hint { font-size: 0.74em; padding: 6px 8px 0; }\n}\n<\/style>\n\n<div id=\"circth_k3Vn_app\">\n    <div class=\"circth_k3Vn_toolbar\">\n        <div class=\"circth_k3Vn_title\">Circle Theorems <span>Mr Barton Maths<\/span><\/div>\n    <\/div>\n\n    <div class=\"circth_k3Vn_tab_bar\" id=\"circth_k3Vn_tabBar\">\n        <button class=\"circth_k3Vn_tab circth_k3Vn_on\" data-theorem=\"centre\" title=\"Angle at the centre is twice the angle at the circumference\">\n            <span class=\"circth_k3Vn_tab_num\">1<\/span>\n            <span class=\"circth_k3Vn_tab_icon\"><svg viewBox=\"0 0 24 24\"><circle cx=\"12\" cy=\"12\" r=\"9.5\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"12\" y1=\"12\" x2=\"5.5\" y2=\"8.5\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"12\" y1=\"12\" x2=\"18.5\" y2=\"8.5\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"12\" y1=\"21.5\" x2=\"5.5\" y2=\"8.5\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"12\" y1=\"21.5\" x2=\"18.5\" y2=\"8.5\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><circle cx=\"12\" cy=\"12\" r=\"1.6\" fill=\"currentColor\"\/><\/svg><\/span>\n            Angle at the Centre\n        <\/button>\n        <button class=\"circth_k3Vn_tab\" data-theorem=\"semicircle\" title=\"Angle in a semicircle is 90\u00b0\">\n            <span class=\"circth_k3Vn_tab_num\">2<\/span>\n            <span class=\"circth_k3Vn_tab_icon\"><svg viewBox=\"0 0 24 24\"><circle cx=\"12\" cy=\"12\" r=\"9.5\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"2.5\" y1=\"12\" x2=\"21.5\" y2=\"12\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"2.5\" y1=\"12\" x2=\"14\" y2=\"3\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"14\" y1=\"3\" x2=\"21.5\" y2=\"12\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><circle cx=\"12\" cy=\"12\" r=\"1.4\" fill=\"currentColor\"\/><\/svg><\/span>\n            Angle in a Semicircle\n        <\/button>\n        <button class=\"circth_k3Vn_tab\" data-theorem=\"same_segment\" title=\"Angles in the same segment are equal\">\n            <span class=\"circth_k3Vn_tab_num\">3<\/span>\n            <span class=\"circth_k3Vn_tab_icon\"><svg viewBox=\"0 0 24 24\"><circle cx=\"12\" cy=\"12\" r=\"9.5\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"6.5\" y1=\"17\" x2=\"5\" y2=\"6\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"6.5\" y1=\"17\" x2=\"19\" y2=\"6\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"17.5\" y1=\"17\" x2=\"5\" y2=\"6\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"17.5\" y1=\"17\" x2=\"19\" y2=\"6\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><\/svg><\/span>\n            Same Segment\n        <\/button>\n        <button class=\"circth_k3Vn_tab\" data-theorem=\"cyclic_quad\" title=\"Opposite angles in a cyclic quadrilateral sum to 180\u00b0\">\n            <span class=\"circth_k3Vn_tab_num\">4<\/span>\n            <span class=\"circth_k3Vn_tab_icon\"><svg viewBox=\"0 0 24 24\"><circle cx=\"12\" cy=\"12\" r=\"9.5\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><polygon points=\"6,7 18,5 20,15 9,20\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\" stroke-linejoin=\"round\"\/><\/svg><\/span>\n            Cyclic Quadrilateral\n        <\/button>\n        <button class=\"circth_k3Vn_tab\" data-theorem=\"tangent_radius\" title=\"A tangent is perpendicular to the radius at the point of contact\">\n            <span class=\"circth_k3Vn_tab_num\">5<\/span>\n            <span class=\"circth_k3Vn_tab_icon\"><svg viewBox=\"0 0 24 24\"><circle cx=\"10\" cy=\"12\" r=\"7.5\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"10\" y1=\"12\" x2=\"17.5\" y2=\"12\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"17.5\" y1=\"3\" x2=\"17.5\" y2=\"21\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><rect x=\"15.3\" y=\"9.7\" width=\"2.2\" height=\"2.2\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.2\"\/><circle cx=\"10\" cy=\"12\" r=\"1.3\" fill=\"currentColor\"\/><\/svg><\/span>\n            Tangent-Radius\n        <\/button>\n        <button class=\"circth_k3Vn_tab\" data-theorem=\"alternate_segment\" title=\"The angle between tangent and chord equals the angle in the alternate segment\">\n            <span class=\"circth_k3Vn_tab_num\">6<\/span>\n            <span class=\"circth_k3Vn_tab_icon\"><svg viewBox=\"0 0 24 24\"><circle cx=\"12\" cy=\"13\" r=\"7.5\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"2\" y1=\"5.5\" x2=\"22\" y2=\"5.5\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"12\" y1=\"5.5\" x2=\"6\" y2=\"18\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"12\" y1=\"5.5\" x2=\"18.5\" y2=\"14\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"6\" y1=\"18\" x2=\"18.5\" y2=\"14\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><\/svg><\/span>\n            Alternate Segment\n        <\/button>\n        <button class=\"circth_k3Vn_tab\" data-theorem=\"two_tangents\" title=\"Tangents from an external point are equal in length\">\n            <span class=\"circth_k3Vn_tab_num\">7<\/span>\n            <span class=\"circth_k3Vn_tab_icon\"><svg viewBox=\"0 0 24 24\"><circle cx=\"9\" cy=\"12\" r=\"6.5\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"20.5\" y1=\"12\" x2=\"6.5\" y2=\"7\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><line x1=\"20.5\" y1=\"12\" x2=\"6.5\" y2=\"17\" stroke=\"currentColor\" stroke-width=\"1.5\"\/><circle cx=\"20.5\" cy=\"12\" r=\"1.2\" fill=\"currentColor\"\/><\/svg><\/span>\n            Two Tangents\n        <\/button>\n    <\/div>\n\n    <div class=\"circth_k3Vn_main\">\n        <div class=\"circth_k3Vn_control_bar\">\n            <div class=\"circth_k3Vn_control_primary\">\n                <button class=\"circth_k3Vn_pill circth_k3Vn_pill_primary\" id=\"circth_k3Vn_revealAnglesBtn\" title=\"Reveal all angle measures\">\u2220 Reveal all angles<\/button>\n            <\/div>\n            <div class=\"circth_k3Vn_control_spacer\"><\/div>\n            <div class=\"circth_k3Vn_control_utilities\">\n                <button class=\"circth_k3Vn_pill circth_k3Vn_pill_secondary circth_k3Vn_on\" id=\"circth_k3Vn_labelsBtn\" title=\"Show or hide point labels A, B, P, O\">A B P<\/button>\n                <div class=\"circth_k3Vn_pill_sep\"><\/div>\n                <button class=\"circth_k3Vn_pill circth_k3Vn_pill_action\" id=\"circth_k3Vn_splitBtn\" title=\"Compare two configurations side by side\">\u25eb Split view<\/button>\n                <button class=\"circth_k3Vn_pill circth_k3Vn_pill_action\" id=\"circth_k3Vn_resetBtn\" title=\"Reset the active board to default positions\">\u21ba Reset<\/button>\n                <button class=\"circth_k3Vn_pill circth_k3Vn_pill_action\" id=\"circth_k3Vn_resetAllBtn\" style=\"display:none;\" title=\"Reset both boards to default positions\">\u21ba Reset all<\/button>\n            <\/div>\n        <\/div>\n\n        <div class=\"circth_k3Vn_boards_row\" id=\"circth_k3Vn_boardsRow\"><\/div>\n\n        <div class=\"circth_k3Vn_tap_hint\">\n            <strong>Tip:<\/strong> tap any shaded angle to reveal its measure. Drag the points A, B, and P around the circle.\n        <\/div>\n\n        <div class=\"circth_k3Vn_theorem_panel\" id=\"circth_k3Vn_theoremPanel\">\n            <button class=\"circth_k3Vn_theorem_toggle\" id=\"circth_k3Vn_theoremToggle\">\n                <div class=\"circth_k3Vn_theorem_toggle_left\">\n                    <div class=\"circth_k3Vn_theorem_toggle_icon\">\ud83d\udca1<\/div>\n                    <div class=\"circth_k3Vn_theorem_toggle_text\">Show the theorem<\/div>\n                <\/div>\n                <span class=\"circth_k3Vn_theorem_chevron\">\u25be<\/span>\n            <\/button>\n            <div class=\"circth_k3Vn_theorem_body\" id=\"circth_k3Vn_theoremBody\">\n                <div class=\"circth_k3Vn_theorem_body_inner\">\n                    <div class=\"circth_k3Vn_theorem_title\">The angle at the centre is twice the angle at the circumference<\/div>\n                    <div class=\"circth_k3Vn_theorem_text\">When both angles are drawn from the <strong>same chord<\/strong>, the angle at the centre is exactly <strong>twice<\/strong> the angle at the circumference.<\/div>\n                    <div class=\"circth_k3Vn_theorem_invite\">Try moving <strong>P<\/strong> around the circle \u2014 does the relationship still hold?<\/div>\n                <\/div>\n            <\/div>\n        <\/div>\n\n        <div class=\"circth_k3Vn_invest_panel\" id=\"circth_k3Vn_investPanel\">\n            <button class=\"circth_k3Vn_invest_toggle\" id=\"circth_k3Vn_investToggle\">\n                <div class=\"circth_k3Vn_invest_toggle_left\">\n                    <div class=\"circth_k3Vn_invest_toggle_icon\">\ud83d\udd0d<\/div>\n                    <div class=\"circth_k3Vn_invest_toggle_text\">Investigations<\/div>\n                <\/div>\n                <span class=\"circth_k3Vn_invest_chevron\">\u25be<\/span>\n            <\/button>\n            <div class=\"circth_k3Vn_invest_body\" id=\"circth_k3Vn_investBody\">\n                <div class=\"circth_k3Vn_invest_body_inner\">\n                    <div class=\"circth_k3Vn_invest_section\" id=\"circth_k3Vn_investPerTheorem\">\n                        <!-- Per-theorem investigation content populated by JS -->\n                    <\/div>\n                    <div class=\"circth_k3Vn_invest_divider\"><\/div>\n                    <div class=\"circth_k3Vn_invest_section\">\n                        <div class=\"circth_k3Vn_invest_section_heading\">\ud83d\udd17 Going Further: Cross-Theorem Investigations<\/div>\n                        <div class=\"circth_k3Vn_invest_section_intro\">Use <strong>Split View<\/strong> to compare two theorems side by side. These investigations reveal deep connections between the seven theorems.<\/div>\n                        <div class=\"circth_k3Vn_invest_q\">\n                            <div class=\"circth_k3Vn_invest_q_num\">A<\/div>\n                            <div class=\"circth_k3Vn_invest_q_body\"><strong>Semicircle as a special case.<\/strong> Open split view. Set the left board to <strong>Theorem 1<\/strong> and the right to <strong>Theorem 2<\/strong>. On the left, drag A and B so the centre angle is exactly 180\u00b0 (a straight line through the centre). What is the angle at the circumference? Compare with the right board. Theorem 2 is what happens when Theorem 1&#8217;s centre angle becomes a straight line \u2014 explain why.<\/div>\n                        <\/div>\n                        <div class=\"circth_k3Vn_invest_q\">\n                            <div class=\"circth_k3Vn_invest_q_num\">B<\/div>\n                            <div class=\"circth_k3Vn_invest_q_body\"><strong>From &#8220;same segment&#8221; to cyclic quadrilateral.<\/strong> Set left to <strong>Theorem 3<\/strong>, right to <strong>Theorem 4<\/strong>. On the left, drag Q across the chord into the opposite segment. The four points A, P, B, Q now form a cyclic quadrilateral. Reveal the angles on the left \u2014 what do they sum to? Compare with the right. The &#8220;supplementary angles in opposite segments&#8221; you saw in Theorem 3 <em>is<\/em> Theorem 4.<\/div>\n                        <\/div>\n                        <div class=\"circth_k3Vn_invest_q\">\n                            <div class=\"circth_k3Vn_invest_q_num\">C<\/div>\n                            <div class=\"circth_k3Vn_invest_q_body\"><strong>Two tangents = Tangent-radius applied twice.<\/strong> Set left to <strong>Theorem 5<\/strong>, right to <strong>Theorem 7<\/strong>. On the right, notice the two right-angle squares at T\u2081 and T\u2082. Each one is Theorem 5! Theorem 7 is just &#8220;Theorem 5 applied at two different points.&#8221; Use this to explain why ET\u2081 = ET\u2082 (hint: triangles OT\u2081E and OT\u2082E share OE and have OT\u2081 = OT\u2082 = r, and both have a right angle \u2014 they&#8217;re congruent).<\/div>\n                        <\/div>\n                        <div class=\"circth_k3Vn_invest_q\">\n                            <div class=\"circth_k3Vn_invest_q_num\">D<\/div>\n                            <div class=\"circth_k3Vn_invest_q_body\"><strong>Alternate segment and angle at the centre.<\/strong> Set left to <strong>Theorem 6<\/strong>, right to <strong>Theorem 1<\/strong>. Look carefully: the tangent at T in Theorem 6 is the &#8220;limit&#8221; of a chord whose other endpoint moves to coincide with T. The angle between the tangent and chord TA is half the central angle subtended by TA. Verify by setting up matching configurations on both boards.<\/div>\n                        <\/div>\n                        <div class=\"circth_k3Vn_invest_q\">\n                            <div class=\"circth_k3Vn_invest_q_num\">E<\/div>\n                            <div class=\"circth_k3Vn_invest_q_body\"><strong>The hidden cyclic quadrilateral in two tangents.<\/strong> Set left to <strong>Theorem 4<\/strong>, right to <strong>Theorem 7<\/strong>. In Theorem 7, reveal the angles at O and E \u2014 they sum to 180\u00b0. The two right angles at T\u2081 and T\u2082 also sum to 180\u00b0. So OT\u2081ET\u2082 has both pairs of opposite angles summing to 180\u00b0 \u2014 which means OT\u2081ET\u2082 is itself a cyclic quadrilateral! What circle is it inscribed in? (Hint: it&#8217;s the circle with diameter OE.)<\/div>\n                        <\/div>\n                    <\/div>\n                <\/div>\n            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Circle Theorems Mr Barton Maths 1 Angle at the Centre 2 Angle in a Semicircle 3 Same Segment 4 Cyclic Quadrilateral 5 Tangent-Radius 6 Alternate Segment 7 Two Tangents \u2220 Reveal all angles A B P \u25eb Split view \u21ba Reset \u21ba Reset all Tip: tap any shaded angle to reveal its measure. 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