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# Triangle and Integer | PRMO 2019 | Question 28

Try this beautiful problem from the Pre-RMO, 2019 based on Circles,Triangle and largest integer.

## Circles, Triangle and largest Integer - PRMO 2019

Let ABC be a triangle with sides 51, 52, 53. Let D denote the incircle of triangle ABC. Draw tangents to D which are parallel to the sides ABC. let $$r_1$$. $$r_2$$, $$r_3$$ be the inradii of the three corner triangles so formed, find the largest integer that does not exceed $$r_1+r_2+r_3$$.

• is 107
• is 15
• is 840
• cannot be determined from the given information

Circles

Triangle

Largest Integers

## Check the Answer

PRMO, 2019, Question 28

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

Let PQ be one of tangents parallel to BC and meet sides AB and AC at P and Q let PQ=x and BC=51

triangle ABC similar with triangle APQ then$$\frac{x}{a}=\frac{r_1}{r}=\frac{s-a}{s}$$ which is in same way for $$\frac{y}{b}$$ and $$\frac{z}{c}$$ then $$\frac{r_1+r_2+r_3}{r}$$=$$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}$$=3-2=1

then $$r_1+r_2+r_3$$=r and r by given condition of question =$$(\frac{s(s-a)(s-b)(s-c)}{s})^\frac{1}{2}$$=$$(\frac{78(78-51)(78-52)(78-53)}{78})^\frac{1}{2}$$=15.

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