Triangle Geometry Solution

Overview

A concise visual proof of a geometry problem involving two right isosceles triangles sharing a common right‑angle vertex, their centroids, perpendiculars to a common base, and the resulting right‑angled trapezoid. The key takeaway is the Pythagorean relationship $(7\sqrt{2})^2+(\sqrt{2})^2=100$ leading to a hypotenuse of 10 cm.

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Phases

#	Phase Name	Duration	Description
1	Scene 1 – Triangle Placement	~5 s	Draw a horizontal base line. On the left, construct right‑isosceles triangle $ABC$ with right angle at $A$, hypotenuse labeled "18 cm" and legs "9\sqrt{2}\,cm". On the right, construct a second right‑isosceles triangle $ADE$ sharing vertex $A$ and the same base line, hypotenuse "24 cm" and legs "12\sqrt{2}\,cm". Ensure the two triangles sit side‑by‑side without overlapping.
2	Scene 2 – Centroids & Perpendiculars	~6 s	Show the medians of each triangle as light dashed lines, mark the centroids $G_1$ and $G_2$ with bright red dots. From each centroid drop a vertical dashed perpendicular to the base: red from $G_1$ meeting the base at $H_1$ and blue from $G_2$ meeting at $H_2$. Label the segment on the red perpendicular "3\sqrt{2}\,cm" (both above and below $G_1$). Label the blue perpendicular "4\sqrt{2}\,cm" (both above and below $G_2$).
3	Scene 3 – Trapezoid and Final Proof	~8 s	Connect $H_1$, $G_1$, $G_2$, $H_2$ to form a right‑angled trapezoid, fill it with a semi‑transparent yellow. From $G_1$ draw a horizontal dashed line parallel to $G_2H_2$ to create a small right triangle inside the trapezoid; label its horizontal side "7\sqrt{2}\,cm" and vertical side "\sqrt{2}\,cm". Draw the segment $G_1G_2$ as a bright white line, overlay the Pythagorean computation $(7\sqrt{2})^2 + (\sqrt{2})^2 = 100$ and highlight the result "10 cm".

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│               MAIN (visual content)         │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional, small) – shows current   │
│  phase title or key equation                 │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	All geometric objects: base line, two triangles, medians, centroids, perpendiculars, trapezoid, inner triangle, hypotenuse line, and all labels.	Uses most of the frame; dark background with bright colors for contrast.
Caption	Small footer displaying the current scene name (e.g., "Scene 1 – Triangle Placement") or the displayed equation in Scene 3.	Optional; appears only when a textual cue aids the viewer.

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Notes

• Assumptions: No axis lines are drawn as requested; the base line serves as the only reference. All lengths are shown as text labels attached to the corresponding segments. Colors: dark background, red for $G_1$ and its perpendicular, blue for $G_2$ and its perpendicular, bright white for $G_1G_2$, semi‑transparent yellow for the trapezoid.
• The animation is in vertical 9:16 format; the main visual is centered vertically with enough margin for the caption band at the bottom.
• Transitions between scenes are simple fades (≈0.5 s) to keep pacing smooth.
• All dashed lines are light (low opacity) to keep focus on solid colored elements.
• No coordinate axes or grid are used, adhering to the user’s restriction.
• Total runtime ≈ 19 seconds, which balances clarity with brevity.