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Omni-MATH / Pick a subset of at least four of the following geometric theorems, order them…

Problem

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problem

Pick a subset of at least four of the following geometric theorems, order them from earliest to latest by publication date, and write down their labels (a single capital letter) in that order. If a theorem was discovered multiple times, use the publication date corresponding to the geometer for which the theorem is named. C. (Ceva) Three cevians AD,BE,CFA D, B E, C F of a triangle ABCA B C are concurrent if and only if BDDCCEEAAFFB=1\frac{B D}{D C} \frac{C E}{E A} \frac{A F}{F B}=1. E. (Euler) In a triangle ABCA B C with incenter II and circumcenter OO, we have IO2=R(R−2r)I O^{2}=R(R-2 r), where rr is the inradius and RR is the circumradius of ABCA B C. H. (Heron) The area of a triangle ABCA B C is s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}, where s=12(a+b+c)s=\frac{1}{2}(a+b+c). M. (Menelaus) If D,E,FD, E, F lie on lines BC,CA,ABB C, C A, A B, then they are collinear if and only if BDDCCEEAAFFB=\frac{B D}{D C} \frac{C E}{E A} \frac{A F}{F B}= -1, where the ratios are directed. P. (Pascal) Intersections of opposite sides of cyclic hexagons are collinear. S. (Stewart) Let ABCA B C be a triangle and DD a point on BCB C. Set m=BD,n=CD,d=ADm=B D, n=C D, d=A D. Then man+dad=bmb+cncm a n+d a d=b m b+c n c V. (Varignon) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. If your answer is a list of 4≤N≤74 \leq N \leq 7 labels in a correct order, your score will be (N−2)(N−3)(N-2)(N-3). Otherwise, your score will be zero.
Plain-text mathematical notation (without MathML)
Pick a subset of at least four of the following geometric theorems, order them from earliest to latest by publication date, and write down their labels (a single capital letter) in that order. If a theorem was discovered multiple times, use the publication date corresponding to the geometer for which the theorem is named. C. (Ceva) Three cevians AD,BE,CF of a triangle ABC are concurrent if and only if (BD)/(DC)(CE)/(EA)(AF)/(FB)=1. E. (Euler) In a triangle ABC with incenter I and circumcenter O, we have IO²=R(R−2r), where r is the inradius and R is the circumradius of ABC. H. (Heron) The area of a triangle ABC is √(s(s−a)(s−b)(s−c)), where s=(1)/(2)(a+b+c). M. (Menelaus) If D,E,F lie on lines BC,CA,AB, then they are collinear if and only if (BD)/(DC)(CE)/(EA)(AF)/(FB)= -1, where the ratios are directed. P. (Pascal) Intersections of opposite sides of cyclic hexagons are collinear. S. (Stewart) Let ABC be a triangle and D a point on BC. Set m=BD,n=CD,d=AD. Then man+dad=bmb+cnc V. (Varignon) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. If your answer is a list of 4≤N≤7 labels in a correct order, your score will be (N−2)(N−3). Otherwise, your score will be zero.
Original LaTeX notation
Pick a subset of at least four of the following geometric theorems, order them from earliest to latest by publication date, and write down their labels (a single capital letter) in that order. If a theorem was discovered multiple times, use the publication date corresponding to the geometer for which the theorem is named. C. (Ceva) Three cevians $A D, B E, C F$ of a triangle $A B C$ are concurrent if and only if $\frac{B D}{D C} \frac{C E}{E A} \frac{A F}{F B}=1$. E. (Euler) In a triangle $A B C$ with incenter $I$ and circumcenter $O$, we have $I O^{2}=R(R-2 r)$, where $r$ is the inradius and $R$ is the circumradius of $A B C$. H. (Heron) The area of a triangle $A B C$ is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{1}{2}(a+b+c)$. M. (Menelaus) If $D, E, F$ lie on lines $B C, C A, A B$, then they are collinear if and only if $\frac{B D}{D C} \frac{C E}{E A} \frac{A F}{F B}=$ -1, where the ratios are directed. P. (Pascal) Intersections of opposite sides of cyclic hexagons are collinear. S. (Stewart) Let $A B C$ be a triangle and $D$ a point on $B C$. Set $m=B D, n=C D, d=A D$. Then $m a n+d a d=b m b+c n c$ V. (Varignon) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. If your answer is a list of $4 \leq N \leq 7$ labels in a correct order, your score will be $(N-2)(N-3)$. Otherwise, your score will be zero.

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Source and history

task_id
69dd5988-8564-566f-bb2b-70194df26102
task_key
test--69dd5988-8564-566f-bb2b-70194df26102
task_revision_id
1
config / split
/ test

Official source

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