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Omni-MATH / Let ABC be a triangle. The following diagram contains points P₁,P₂,…,P₇, which are the following triangle centers of triangle ABC in some order: - the incenter I; - the circumcente…

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problem

Let ABCA B C be a triangle. The following diagram contains points P1,P2,…,P7P_{1}, P_{2}, \ldots, P_{7}, which are the following triangle centers of triangle ABCA B C in some order: - the incenter II; - the circumcenter OO; - the orthocenter HH; - the symmedian point LL, which is the intersections of the reflections of BB-median and CC-median across angle bisectors of ∠ABC\angle A B C and ∠ACB\angle A C B, respectively; - the Gergonne point GG, which is the intersection of lines from BB and CC to the tangency points of the incircle with AC¯\overline{A C} and AB¯\overline{A B}, respectively; - the Nagel point NN, which is the intersection of line from BB to the tangency point between BB excircle and AC¯\overline{A C}, and line from CC to the tangency point between CC-excircle and AB¯\overline{A B}; and - the Kosnita point KK, which is the intersection of lines from BB and CC to the circumcenters of triangles AOCA O C and AOBA O B, respectively. Compute which triangle centers {I,O,H,L,G,N,K}\{I, O, H, L, G, N, K\} corresponds to PkP_{k} for k∈{1,2,3,4,5,6,7}k \in\{1,2,3,4,5,6,7\}.
Plain-text mathematical notation (without MathML)
Let ABC be a triangle. The following diagram contains points P₁,P₂,…,P₇, which are the following triangle centers of triangle ABC in some order: - the incenter I; - the circumcenter O; - the orthocenter H; - the symmedian point L, which is the intersections of the reflections of B-median and C-median across angle bisectors of ∠ABC and ∠ACB, respectively; - the Gergonne point G, which is the intersection of lines from B and C to the tangency points of the incircle with (AC)¯ and (AB)¯, respectively; - the Nagel point N, which is the intersection of line from B to the tangency point between B excircle and (AC)¯, and line from C to the tangency point between C-excircle and (AB)¯; and - the Kosnita point K, which is the intersection of lines from B and C to the circumcenters of triangles AOC and AOB, respectively. Compute which triangle centers {I,O,H,L,G,N,K} corresponds to P_(k) for k∈{1,2,3,4,5,6,7}.
Original LaTeX notation
Let $A B C$ be a triangle. The following diagram contains points $P_{1}, P_{2}, \ldots, P_{7}$, which are the following triangle centers of triangle $A B C$ in some order: - the incenter $I$; - the circumcenter $O$; - the orthocenter $H$; - the symmedian point $L$, which is the intersections of the reflections of $B$-median and $C$-median across angle bisectors of $\angle A B C$ and $\angle A C B$, respectively; - the Gergonne point $G$, which is the intersection of lines from $B$ and $C$ to the tangency points of the incircle with $\overline{A C}$ and $\overline{A B}$, respectively; - the Nagel point $N$, which is the intersection of line from $B$ to the tangency point between $B$ excircle and $\overline{A C}$, and line from $C$ to the tangency point between $C$-excircle and $\overline{A B}$; and - the Kosnita point $K$, which is the intersection of lines from $B$ and $C$ to the circumcenters of triangles $A O C$ and $A O B$, respectively. Compute which triangle centers $\{I, O, H, L, G, N, K\}$ corresponds to $P_{k}$ for $k \in\{1,2,3,4,5,6,7\}$.

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task_id
e6284b57-fb45-5af2-a7f8-4f6804681c1a
task_key
test--e6284b57-fb45-5af2-a7f8-4f6804681c1a
task_revision_id
1
config / split
/ test

Official source

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