If we have a square ABCD and a rectangle AEFD. Let P be the intersection of AC and ED, Q the intersection of AF and BD. How can we prove that the line PQ is parallel to AD?
Hello, geo_math!
We have a square and a rectangle .
Let be the intersection of and , the intersection of and
Prove that is parallel toCode:Q * : * * : * * : * * : * * : * θ * A : B * - - - - - * - - - - - - - - - - - * E : | * α * | * θ θ * | : | * * α | * * | : | * | * | : | * R * α | * * | : | * α * | * θ θ * | : C * - - - - - * - - - - - - - - - - - * F : * θ * D : * * : * * : * * : * * P *
Let and intersect at
Note that all angles labeled are equal.
. . And all angles labeled are 45°.
Further note that: .
In right triangles and
Hence: .
Then is an isosceles right triangle: .
Since