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Geometry Level 4

In acute angled triangle ABC , c > a > b c>a>b the line I O IO

( I : i n c e n t r e ; O : c i r c u m c e n t r e I: incentre; O:circumcentre ) intersects the line segment.. (where a , b , c a,b,c are length of sides of triangle ABC)

A C AC and B C BC N o n e None A B AB and B C BC A C AC and A B AB

Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

Since a right angled triangle is the limiting case of an acute angled triangle, it will give us an indication.
In right angle triangle, O is on biggest side. While I is towards the smallest side from O.
If we let b be very small this is very clear. The problem implies that all acute angled triangle have the same behavior. OR.
Since a right angled triangle is the limiting case of an acute angled triangle, it will give us an indication.
I is on angle bisector of the right angle intercecting c at D. O is on c.
(We know that for an acute angled triangle O is in side the triangle. I is inside for all triangles.)
We see that CD is between CO and the smaller side CA.
So any line drawn from O to a point on CD will intersect the small side CA. I is on CD.
So OI intersect bigger side c and small b.

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