Theorem 15.1.1

Projection of a side of Triangle: In this chapter, we will be studying some basic and famous theorems related to the projection of a side of a triangle on another side.

Theorem 15.1.1: In an obtuse-angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing the obtuse angle together with twice the rectangle contained by one of the sides, and the projection on it of the other.

Given: $\triangle ABC$ with an obtuse angle at vertex A.

To prove: $a^2=b^2+c^2+2cx$

Construction: Draw a perpendicular line $\overline{CD}$ on $\overline{BA}$ produced, meeting at point D, so that $\overline{AD}$ is the projection of $\overline{AC}$ on $\overline{BA}$ produced. Taking $m\overline{BC} = a$, $m\overline{CA} = b$, $m\overline{BA} = c$, $m\overline{AD} = x$, and $m\overline{CD} = h$.

Proof: