In a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, let
![M_{a}](/media/m/4/0/0/40084e8314625e88c07254a23638cb3f.png)
,
![M_{b}](/media/m/9/b/2/9b22584803aa23e6da2748b31077c3a4.png)
,
![M_{c}](/media/m/7/4/9/7492a745791f6313c924704a571a315a.png)
be the midpoints of the sides
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, respectively, and
![T_{a}](/media/m/a/3/8/a38153c6ceb82e0e690bd4fa7c00adfb.png)
,
![T_{b}](/media/m/2/4/b/24b36656abf6b176a52db53d7ada1b13.png)
,
![T_{c}](/media/m/b/6/0/b60de343b788a82aeb9ffdd509c5785e.png)
be the midpoints of the arcs
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
of the circumcircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, not containing the vertices
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
, respectively. For
![i \in \left\{a, b, c\right\}](/media/m/c/2/1/c219c2c7bd7d5a0d0e6869e6d8d42756.png)
, let
![w_{i}](/media/m/e/8/0/e80e3cdecb39614db6760dfc4e89009c.png)
be the circle with
![M_{i}T_{i}](/media/m/c/3/0/c30cc11daee535cb24546f3a65ff6038.png)
as diameter. Let
![p_{i}](/media/m/7/8/c/78cd12e5ab60059f6fe6f6d8a5a706a3.png)
be the common external common tangent to the circles
![w_{j}](/media/m/9/8/1/9812d4a3704ae4bed619df04f052c9fc.png)
and
![w_{k}](/media/m/1/4/3/14307975cb6908590d9f09f6e784efd3.png)
(for all
![\left\{i, j, k\right\}= \left\{a, b, c\right\}](/media/m/2/e/7/2e73a9f2036acda81462ba6f22eff454.png)
) such that
![w_{i}](/media/m/e/8/0/e80e3cdecb39614db6760dfc4e89009c.png)
lies on the opposite side of
![p_{i}](/media/m/7/8/c/78cd12e5ab60059f6fe6f6d8a5a706a3.png)
than
![w_{j}](/media/m/9/8/1/9812d4a3704ae4bed619df04f052c9fc.png)
and
![w_{k}](/media/m/1/4/3/14307975cb6908590d9f09f6e784efd3.png)
do.
Prove that the lines
![p_{a}](/media/m/e/3/4/e34576a7b4f6bb4f7b9dc224f9f49bcc.png)
,
![p_{b}](/media/m/e/6/0/e605b32d71a9153e1fc17a0e2de31f41.png)
,
![p_{c}](/media/m/3/f/e/3fe98c21d8795b59168ed9274a6ae157.png)
form a triangle similar to
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and find the ratio of similitude.
%V0
In a triangle $ABC$, let $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively, and $T_{a}$, $T_{b}$, $T_{c}$ be the midpoints of the arcs $BC$, $CA$, $AB$ of the circumcircle of $ABC$, not containing the vertices $A$, $B$, $C$, respectively. For $i \in \left\{a, b, c\right\}$, let $w_{i}$ be the circle with $M_{i}T_{i}$ as diameter. Let $p_{i}$ be the common external common tangent to the circles $w_{j}$ and $w_{k}$ (for all $\left\{i, j, k\right\}= \left\{a, b, c\right\}$) such that $w_{i}$ lies on the opposite side of $p_{i}$ than $w_{j}$ and $w_{k}$ do.
Prove that the lines $p_{a}$, $p_{b}$, $p_{c}$ form a triangle similar to $ABC$ and find the ratio of similitude.