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Omni-MATH / In acute triangle ABC, let D,E, and F be the feet of the altitudes from A,B, and C respectively, and let L,M, and N be the midpoints of BC,CA, and AB, respectively. Lines DE and NL…

Problem

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problem

In acute triangle ABCA B C, let D,ED, E, and FF be the feet of the altitudes from A,BA, B, and CC respectively, and let L,ML, M, and NN be the midpoints of BC,CAB C, C A, and ABA B, respectively. Lines DED E and NLN L intersect at XX, lines DFD F and LML M intersect at YY, and lines XYX Y and BCB C intersect at ZZ. Find ZBZC\frac{Z B}{Z C} in terms of AB,ACA B, A C, and BCB C.
Plain-text mathematical notation (without MathML)
In acute triangle ABC, let D,E, and F be the feet of the altitudes from A,B, and C respectively, and let L,M, and N be the midpoints of BC,CA, and AB, respectively. Lines DE and NL intersect at X, lines DF and LM intersect at Y, and lines XY and BC intersect at Z. Find (ZB)/(ZC) in terms of AB,AC, and BC.
Original LaTeX notation
In acute triangle $A B C$, let $D, E$, and $F$ be the feet of the altitudes from $A, B$, and $C$ respectively, and let $L, M$, and $N$ be the midpoints of $B C, C A$, and $A B$, respectively. Lines $D E$ and $N L$ intersect at $X$, lines $D F$ and $L M$ intersect at $Y$, and lines $X Y$ and $B C$ intersect at $Z$. Find $\frac{Z B}{Z C}$ in terms of $A B, A C$, and $B C$.

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Source and history

task_id
9ab06cfb-c34d-54ef-b596-d2d331ce80c8
task_key
test--9ab06cfb-c34d-54ef-b596-d2d331ce80c8
task_revision_id
1
config / split
/ test

Official source

initial import