![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
is a point on the semicircle diameter
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, between
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
.
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
is the foot of the perpendicular from
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
. The circle
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
is the incircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, the circle
![K_2](/media/m/3/1/d/31d4e457d9fd74913d14bd19a565ce00.png)
touches
![CD,DA](/media/m/8/5/0/850f70fc9d766d76909fc6e3571530c3.png)
and the semicircle, the circle
![K_3](/media/m/d/7/c/d7c055636fbf1f3b807bc72796996893.png)
touches
![CD,DB](/media/m/a/a/b/aab5721dd068a7669a10449990e22f7b.png)
and the semicircle. Prove that
![K_1,K_2](/media/m/7/5/1/75111418db9478ff7ca80ae9fefb8265.png)
and
![K_3](/media/m/d/7/c/d7c055636fbf1f3b807bc72796996893.png)
have another common tangent apart from
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
.
%V0
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.