Let points divide the side of into parts

MATHEMATICS · Subjective

Problem

Let points

P_1,P_2,P_3,\dots,P_{n-1}

divide the side

BC

\triangle ABC

into

n

parts. Let

r_1,r_2,r_3,\dots,r_n

be the radii of the incircles and let

p_1,p_2,\dots,p_n

be the radii of the excircles corresponding to vertex

A

for triangles

ABP_1

AP_1P_2

\dots

AP_{n-1}C

. Let

r

and

p

be the corresponding radii for triangle

ABC

. Show that

\frac{r_1}{p_1}\cdot\frac{r_2}{p_2}\cdots\frac{r_n}{p_n}=\frac{r}{p}