Law of Cosines → Stewart's Theorem

Overview

A concise visual proof of the Law of Cosines followed by its application to derive Stewart's Theorem for a cevian in a triangle. Viewers see how the algebraic manipulation of two cosine relations yields the classic formula
$$d^{2}=\frac{c^{2}n+b^{2}m}{m+n}-mn$$.
The key takeaway: Stewart’s theorem is a direct consequence of the Law of Cosines applied to the two sub‑triangles created by a cevian.

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Phases

#	Phase Name	Duration	Description
1	Intro	~5 s	Title appears: “Law of Cosines → Stewart’s Theorem”. A simple triangle outline fades in, establishing the setting.
2	Law of Cosines Proof	~20 s	Triangle $\triangle ABC$ is drawn. Angle $\angle C$ is highlighted; side $c$ opposite $C$ is labeled. A perpendicular from $C$ to extension of $AB$ creates two right‑triangles. Squares on each side appear, and the classic rearrangement shows $c^{2}=a^{2}+b^{2}-2ab\cos C$.
3	Set‑up for Stewart	~15 s	Point $D$ is placed on side $BC$. Segments $m=BD$ and $n=DC$ are marked; cevian $AD$ (length $d$) is drawn. The two sub‑triangles $\triangle ABD$ and $\triangle ACD$ are highlighted in alternating colors.
4	Derivation using Cosines	~15 s	Apply the Law of Cosines to $\triangle ABD$: $b^{2}=m^{2}+d^{2}-2md\cos\angle ABD$. Then to $\triangle ACD$: $a^{2}=n^{2}+d^{2}-2nd\cos\angle ACD$. Because $\angle ABD$ and $\angle ACD$ are supplementary, $\cos\angle ABD = -\cos\angle ACD$. Adding the two equations eliminates the cosine term, yielding the Stewart relation. The algebraic steps appear as brief on‑screen transformations (no spoken text).
5	Final Formula & Highlight	~4 s	The final compact form $$d^{2}=\frac{c^{2}n+b^{2}m}{m+n}-mn$$ fades in at the bottom caption area, while the full triangle remains static. A quick highlight circles the terms to reinforce the result.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│               MAIN (triangle & shapes)      │
│                                             │
├─────────────────────────────────────────────┤
│ Caption / key formula (persistent footer)   │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Dynamic triangle $ABC$, point $D$, cevian $AD$, right‑triangle constructions, colored sub‑triangles, algebraic symbols appearing near the relevant sides.	Uses two alternating highlight colors for sub‑triangles; minimal persistent text.
Caption	Current key equation (e.g., Law of Cosines, then Stewart’s formula) displayed throughout the relevant phase.	Updated at phase boundaries; stays in the lower band for easy reading on a short‑form (vertical) format.

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Notes

• Variables chosen for clarity: $a=BC$, $b=CA$, $c=AB$; cevian $AD=d$; segments $m=BD$, $n=DC$. These are standard in textbooks and avoid confusion.
• Vertical orientation assumed for YouTube Shorts; the layout diagram reflects a tall frame (main visual occupies ~85 % height, caption ~15 %).
• No spoken narration; all reasoning is conveyed visually through shapes, arrows, and brief on‑screen equations.
• Timing is approximate; each phase includes smooth transitions (fade‑in/out, color shifts) that keep the total under 59 seconds.
• Assumptions made due to lack of explicit style requests:
  1. Use of standard Euclidean geometry (no coordinate axes).
  2. Emphasis on visual algebraic manipulation rather than detailed textual derivation.
  3. Colors: primary triangle in white, sub‑triangles in teal and orange, caption in light gray background for readability.