Given a point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
inside a triangle
![\triangle ABC](/media/m/1/f/3/1f3c3c0f3e134a169655f9511ba6ea82.png)
. Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
be the orthogonal projections of the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
on 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. Let the orthogonal projections of the point
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
on the lines
![BP](/media/m/e/e/f/eefb4fe46ab8d85b7067c29b24aa4cfc.png)
and
![CP](/media/m/6/3/0/630424587cadeb75669118dab3df6b98.png)
be
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
, respectively. Prove that the lines
![ME](/media/m/0/9/f/09fe7f54e93fe547123cecfc182bc6e2.png)
,
![NF](/media/m/d/8/c/d8ce759c605821cbb17d88434f932077.png)
,
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
are concurrent.
Original formulation:
Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be any triangle and
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
any point in its interior. Let
![P_1, P_2](/media/m/6/6/5/665af52c420a67f1d459d47268037345.png)
be the feet of the perpendiculars from
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
to the two sides
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![BC.](/media/m/c/8/0/c808a4699a08ca37e026a4f20c00723c.png)
Draw
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
and
![BP,](/media/m/f/5/5/f55d37eee6d4a77b1cc936fe4e02a080.png)
and from
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
drop perpendiculars to
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
and
![BP.](/media/m/f/c/8/fc8bc3044cee9e643b2307ad6f3389f1.png)
Let
![Q_1](/media/m/0/3/1/0313e7ec2e52d7e7514e810cd41daf66.png)
and
![Q_2](/media/m/7/5/f/75f4681412f8aab76f96fc5b7786365f.png)
be the feet of these perpendiculars. Prove that the lines
![Q_1P_2,Q_2P_1,](/media/m/7/0/c/70ca2a336b6f3c9becf4cbcacd87e6fb.png)
and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
are concurrent.
%V0
Given a point $P$ inside a triangle $\triangle ABC$. Let $D$, $E$, $F$ be the orthogonal projections of the point $P$ on the sides $BC$, $CA$, $AB$, respectively. Let the orthogonal projections of the point $A$ on the lines $BP$ and $CP$ be $M$ and $N$, respectively. Prove that the lines $ME$, $NF$, $BC$ are concurrent.
Original formulation:
Let $ABC$ be any triangle and $P$ any point in its interior. Let $P_1, P_2$ be the feet of the perpendiculars from $P$ to the two sides $AC$ and $BC.$ Draw $AP$ and $BP,$ and from $C$ drop perpendiculars to $AP$ and $BP.$ Let $Q_1$ and $Q_2$ be the feet of these perpendiculars. Prove that the lines $Q_1P_2,Q_2P_1,$ and $AB$ are concurrent.