Co-Side Theorem (共邊定理)

 and shares a common side . Line intersects line at . This theorem says that . There're 4 possible cases depending on how the 2 pairs of points and are divided by the 2 lines and .
\   same sidediff side
same side(a)(b)
diff side(c)(d)
The usual proof involves simple arithmetic manipulations on fractions. This activity aims at providing a constructive proof without words that works for all cases. The principle is to transform into a triangle with equal area and a base , then a similar construction for .

Click either one of the unchecked and move point A' and observe the transformed triangles.

3 proofs