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Estonia — Open Contests

Geometry Difficulty 5.2 AIME, harder Prove it Estonia

Inside a circle $c$ with the center $O$ there are two circles $c_1$ and $c_2$ which go through $O$ and are tangent to the circle $c$ at points $A$ and $B$ respectively. Prove that the circles $c_1$ and $c_2$ have a common point which lies in the segment $AB$ .

Solution

The radius $AO$ of the circle $c$ is perpendicular to the common tangent to circles $c$ and $c_1$ at the point $A$ , hence $AO$ is a diameter of the circle $c_1$ . Similarly $BO$ is a diameter of the circle $c_2$ .

If the circles $c_1$ and $c_2$ are tangent at the point $O$ (Fig. 1), then the diameters $AO$ and $BO$ are both perpendicular to the common tangent to $c_1$ and $c_2$ at the point $O$ , whence the lines $AO$ and $BO$ coincide, i.e. $O$ lies in the segment $AB$ .

If the circles $c_1$ and $c_2$ intersect at $O$ (Fig. 2), then let $M$ be the other intersection point of the circles. Since $\angle AMO = 90^\circ$ and $\angle BMO = 90^\circ$ (angles at the circumference supported by a diameter), the lines $AM$ and $BM$ coincide and $M$ lies in the segment $AB$ .

Figure 1

Figure 2

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