Name: Areas of Similar Triangles
Uploaded: 2019-07-19
Duration: 6 min 54 s
Description: The corresponding sides of similar triangles are proportional. Are the ratios of areas of similar triangles the same as that of their sides? Watch this video...

Question

Three congruent circles have a common point

O

and lie inside a triangle such that each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the point

O

are collinear.

Accepted Answer

Let $K, L, M$ be the centres of the three circles of equal radii, meeting in a common point $O$ , and pairwise touching some side of the triangle $A B C$ in which they lie. Since the three circles have equal radii, we see that $O$ is equidistant from the points $K, L, M$ and so is the circumcentre of the triangle $K L M$ . Also for the same reason, the sides of the triangle $K L M$ are parallel to the corresponding sides of the triangle $A B C$ . (For instance, $K L$ is parallel to $A B$ , as $K$ and $L$ are equidistant from $A B$ )

Further $A K, B L, C M$ are the bisectors of angles $A, B, C$ of triangle $A B C$ respectively. So not only they meet in $I$ , the incentre of the triangle $A B C$ but also $K I, L I, M I$ are the bisectors of the angles of triangle $K L M$ , implying that $I$ is also the incentre of triangle $K L M$ .

It follows, from the relation $\frac{I K}{I A} = \frac{I L}{I B} = \frac{I M}{I C}$ that triangle $K L M$ is homothetic to triangle $A B C$ with respect to $I$ , the centre of homothety. (This simply means that triangle $A B C$ is a dilation (or an enlargement) of triangle $K L M$ as seen from $I$ ).

A property of homothety is that the centre of homothety and any two corresponding points of the homothetic figures are collinear. Here, in particular, $I$ and the circumcentres of triangles $K L M$ and $A B C$ have to be collinear. This is precisely what was to be proved.

Three congruent circles have a common point $O$ and lie inside a triangle such that each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the point $O$ are collinear.

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