Easy proof of the intercept theorem.

Question

Im looking for a simple proof of the Intercept-Theorem in the Euclidean Plane $\mathbb{R}^2$. I can use analytic and synthetic Proofs and Theorems but students should be able to understand it. I've found long proofs with constructions but without use of other theorems.

Answer 1

We can see that $\large{\frac{A_{\triangle{ACE}}}{A_{\triangle{CDE}}}=\frac{AC}{CD}}$ (same height). Similarly, $\large{\frac{A_{\triangle{BCD}}}{A_{\triangle{CDE}}}=\frac{BC}{CE}}$.
Next, $A_{\triangle{ADE}}=A_{\triangle{BDE}}$ (area of two triangles with the same base between parallel lines).
Hence, $A_{\triangle{ACE}}=A_{\triangle{BCD}}$ and $\frac{A_{\triangle{ACE}}}{A_{\triangle{CDE}}}=\frac{A_{\triangle{BCD}}}{A_{\triangle{CDE}}}$
By transitivity, $$\frac{AC}{CD}=\frac{BC}{CE}$$