The incenter of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is
![K.](/media/m/8/7/7/87793463e1777a48f69b354ba5cd56ee.png)
The midpoint of
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
is
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
and that of
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
is
![B_1.](/media/m/0/5/f/05fcb8a49feecb068dc025cf5d3ad704.png)
The lines
![C_1K](/media/m/2/f/6/2f6e2f8c96c2384859798e6f6aaafd99.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
meet at
![B_2,](/media/m/1/d/f/1df34e54de6a0a6d580a405b4c8fb9c5.png)
the lines
![B_1K](/media/m/6/d/b/6db4f25a6d17bf47922cc06e1272ede6.png)
and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
at
![C_2.](/media/m/7/3/8/738aa672284598270d3725a73556f478.png)
If the areas of the triangles
![AB_2C_2](/media/m/c/f/c/cfce614fc4a7bcdb087f16de20441417.png)
and
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
are equal, what is the measure of angle
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The incenter of the triangle $ABC$ is $K.$ The midpoint of $AB$ is $C_1$ and that of $AC$ is $B_1.$ The lines $C_1K$ and $AC$ meet at $B_2,$ the lines $B_1K$ and $AB$ at $C_2.$ If the areas of the triangles $AB_2C_2$ and $ABC$ are equal, what is the measure of angle $\angle CAB?$