Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute-angled triangle. Let
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
be a point such
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
are on distinct sides of the line
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
, and
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
is an interior point of segment
![AE](/media/m/c/e/3/ce31f42a92358c211bccb23e6a92fb55.png)
. We have
![\angle ADB = \angle CDE](/media/m/e/b/1/eb10c97c597c44f1e9086343cb677aff.png)
,
![\angle BAD = \angle ECD](/media/m/a/4/a/a4ad318f5e63bf19f33a74efca04ea64.png)
, and
![\angle ACB = \angle EBA](/media/m/f/c/b/fcbd71a879e158e4adb0136cd6a443f9.png)
. Prove that
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
lie on the same line.
%V0
Let $ABC$ be an acute-angled triangle. Let $E$ be a point such $E$ and $B$ are on distinct sides of the line $AC$, and $D$ is an interior point of segment $AE$. We have $\angle ADB = \angle CDE$, $\angle BAD = \angle ECD$, and $\angle ACB = \angle EBA$. Prove that $B$, $C$ and $E$ lie on the same line.