Answer to Question #302217 in Geometry for salomo

Question

Qn 5. In a scalene triangle whose sides have lengths a, b and c, consider

the bisector r of the angle formed by a and b. Compute, providing your

working based on Euclidean geometry, the ratio between the lengths of the

two segments that the bisector r determines on the side of length c when

intersecting it.

Answer 1

Let ABC be the triangle. Here AB = c, AC = b, BC = a. We look at the bisector of the angle C, let us call this bisector CM = r. We should determine the ratio AM:MB.

Let us denote the half of angle C as $\alpha.$ So the area of triangle ACM is $\frac12 AC\cdot CM\cdot \sin\alpha$ and area of triangle BCM is $\frac12 BC\cdot CM\cdot \sin\alpha$ . Therefore, the ratio of areas is AC:BC. But if we write the formula for area using the height and side, we'll obtain the area of ACM to be $\frac12\cdot AM\cdot h,$ area of BCM to be $\frac12\cdot BM\cdot h,$ so the ratio of areas is AM:BM.

Finally, we obtain AC:BC = AM:BM, or b:a = AM:BM. We also know that AM+BM = c, so $AM = c\cdot\frac{b}{a+b}, BM = c\cdot\frac{a}{a+b}$ .