What theorem's proof is being dynamically illustrated here?[br]And how does this applet actually informally show this theorem to be true? [br](Feel free to move the white points [i]A[/i], [i]B[/i], [i]C[/i] anywhere you'd like.)
[b][color=#000000]Theorem:[/color][/b][br]An angle bisector of a triangle splits the side (opposite that angle) into 2 segments whose lengths are in proportion to the other two sides of the triangle. (That is, in the applet above, [math]\frac{b}{a}=\frac{c_1}{c_2}[/math]). [br][br][b][color=#000000]Here's how: [/color][/b][br]1) The two unlabeled sides are parallel (since [color=#ff00ff][b]alternate interior angles congruent [/b][/color]causes lines to be parallel.) [br]2) Given this and the [url=https://www.geogebra.org/m/NjQaYTCV?doneurl=%2Ftbrzezinski#material/vegmfnfu]Parallel Lines Proportionality Theorem[/url], we can write [math]\frac{c_1}{c_2}=\frac{b}{d}[/math]. [br]3) Yet since [color=#ff00ff][b]angle [i]CBE[/i][/b][/color] and [color=#ff00ff][b]angle [i]CEB[/i][/b][/color] are congruent, we can say [i]a[/i] = [i]d[/i] by the converse of the Isosceles Triangle Theorem. [br][br]4) Therefore, by simple substitution, we can rewrite the equation in (2) as [math]\frac{c_1}{c_2}=\frac{b}{a}[/math].