Square between circles

Geometry Level 4

Given the circle with center at L L , M R S T MRST quadrilateral with vertices on the circle L L , and a circle O O inscribed in the quadrilateral, such that

R M ‾ = 17 , M T ‾ = 19 , T S ‾ = 23 \overline { RM } =17, \ \overline { MT } =19, \ \overline { TS } = 23

What is the value of R T ‾ × M S ‾ \overline{RT} \times \overline{MS} ?


The answer is 790.

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Ivan Molina N by Ivan Molina N , 28, Colombia

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3 solutions

Krishna Sharma
Mar 8, 2015

Since a circle can be inscribed inside a quadrilateral then by pitot's theorem sum of opposite sides are equal

MR + TS = MT + RS

From here we get RS = 21.

Using another property of cyclic quadrilateral

R M × T S + M T × R S = R T × M S \displaystyle RM \times TS + MT \times RS =RT\times MS

From here R T × M S = 790 \displaystyle \boxed{RT \times MS = 790} .

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Ashutosh Kumar
Mar 8, 2015

First apply Pitot's theorem and then Ptolemy's theorem.. Question is Solved

2 Helpful 0 Interesting 0 Brilliant 1 Confused
Raven Herd
Mar 19, 2015

Did just the same way .Guess I rediscovered the pitot's theorem .

0 Helpful 0 Interesting 0 Brilliant 1 Confused

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