Right Triangle Altitude Theorem

Today we will learn about Right Triangle Altitude Theorem and use it to derive Pythagorean Theorem.

Divide a quadrilateral into equal areas

Today let us consider the following interesting compass-and-straightedge construction problem:

Dividing quadrilateral problem. Given a quadrilateral $ABCD$ and four points $M_1$, $M_2$, $M_3$, $M_4$, in this order, on $AB$. Using compass and straightedge, show how to construct four points $N_1$, $N_2$, $N_3$, $N_4$ on $CD$ so that the four line segments $M_1 N_1$, $M_2 N_2$, $M_3 N_3$ and $M_4 N_4$ divide the quadrilateral into 5 small quadrilaterals of equal area.

Similar triangles

Today, we are going to learn about similar triangles. We will use similar triangles to give a proof of Pythagorean Theorem. 

Morley's Theorem

In a triangle $ABC$, draw the trisectors of the angles $A$, $B$, $C$. These trisectors intersect at $A'$, $B'$ and $C'$ as in the figure below. Prove that the triangle $A'B'C'$ is an equilateral triangle.