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In triangle 
, prove that
![\[ a = b\cos C + c\cos B. \]](//latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
, prove that![\[ a = b\cos C + c\cos B. \]](http://latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
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Concurrence, Isogonality
Wictro 40
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by CatinoBarbaraCombinatoric
Source: BMO 2019, Problem 3
Let be an acute scalene triangle. Let 
and 
be two distinct interior points of the segment 
such that 
. Suppose that:
and 
are the feet of the perpendiculars from from 
to the lines 
and 
respectively.
and 
are the feet of the perpendiculars from 
to the lines and respectively.
Prove that
and 
intersect on the line .
be an acute scalene triangle. Let 
and 
be two distinct interior points of the segment 
such that 
. Suppose that:
and 
are the feet of the perpendiculars from from 
to the lines 
and 
respectively.
and 
are the feet of the perpendiculars from 
to the lines and respectively.Prove that
and 
intersect on the line .
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A Projection Theorem
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Source: VN Math Olympiad For High School Students P1 - 2025
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Y by buratinogigle
too lazy to pull up a diagram
but you consider the cases where the triangle is acute, and where either angle B or angle C is obtuse
if the triangle is acute, dropping an altitude from A basically directly shows it
if the triangle is obtuse (let's say C is, other case basically the same), let the intersection of the altitude from A and the line BC be D, BC=c*cos(B), CD=b*cos(pi-C)=-b*cos(C), a=BC=BD-CD=c*cos(B)-(-b*cos(C))=c*cos(B)+b*cos(C)
but you consider the cases where the triangle is acute, and where either angle B or angle C is obtuse
if the triangle is acute, dropping an altitude from A basically directly shows it
if the triangle is obtuse (let's say C is, other case basically the same), let the intersection of the altitude from A and the line BC be D, BC=c*cos(B), CD=b*cos(pi-C)=-b*cos(C), a=BC=BD-CD=c*cos(B)-(-b*cos(C))=c*cos(B)+b*cos(C)
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