334. 2011 Japan Mathematical Olympiad Finals Problem 1.
by Virgil Nicula, Jan 28, 2012, 10:41 PM
PP1. Given an acute triangle 
with the orthocenter 
and the midpoint 
of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Denote the projection 
of on 
. Show that 
.
Proof 1 (metric). Denote
. Using the power of 
w.r.t. the cyclical quadrilateral 
obtain that 
![$m_a^2-bc\cdot \cos A=\frac 14\cdot\left[2\left(b^2+c^2\right)-a^2-2\cdot \left(b^2+c^2-a^2\right)\right]=\frac {a^2}{4}=MB^2$](//latex.artofproblemsolving.com/c/b/f/cbfe99d1a20a9a2c70155143c5e35db74ecac4d0.png)
.
Proof 2 (synthetic). Let
, 
, 
be the projections of to the sides of 
and 
. Are well-known
that
and 
is a harmonical division. In conclusion, 
, i.e 
.
Proof 3 (syntheric). Denote , 
and 
. Observe that 
is cyclically 
and
is cyclically 
.
Proof 4 (proiective). Denote
. Observe that 
, 
and 
are cyclically and their
radical axis
, 
, 
concur at 
, i.e. 
. Since 
is inscribed in the circle with the diameter ![$[AT]$](//latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
obtain that
. Since 
is a harmonical division obtain that 
. In conclusion, 
.
Proof 5 (synthetic). Prove easily that
is inscribed in 
with diameter ![$[AH]$](//latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
and 
is tangent to in 
because 
, i.e. 
. Hence from the power of w.r.t. obtain that 
.
with the orthocenter 
and the midpoint 
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Denote the projection 
of on 
. Show that 
.Proof 1 (metric). Denote
. Using the power of 
w.r.t. the cyclical quadrilateral 
obtain that 
![$m_a^2-bc\cdot \cos A=\frac 14\cdot\left[2\left(b^2+c^2\right)-a^2-2\cdot \left(b^2+c^2-a^2\right)\right]=\frac {a^2}{4}=MB^2$](http://latex.artofproblemsolving.com/c/b/f/cbfe99d1a20a9a2c70155143c5e35db74ecac4d0.png)
.Proof 2 (synthetic). Let
, 
, 
be the projections of to the sides of 
and 
. Are well-knownthat
and 
is a harmonical division. In conclusion, 
, i.e 
.Proof 3 (syntheric). Denote
, 
and 
. Observe that 
is cyclically 
and
is cyclically 
.Proof 4 (proiective). Denote
. Observe that 
, 
and 
are cyclically and theirradical axis
, 
, 
concur at 
, i.e. 
. Since 
is inscribed in the circle with the diameter ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
obtain that
. Since 
is a harmonical division obtain that 
. In conclusion, 
.Proof 5 (synthetic). Prove easily that
is inscribed in 
with diameter ![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
and 
is tangent to in 
because 
, i.e. 
. Hence from the power of w.r.t. obtain that 
. This post has been edited 22 times. Last edited by Virgil Nicula, Nov 19, 2015, 12:37 PM
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