Dao's blog

Let $ABCD$ be a parallelogram. $IG//AB//CD$, $FH//BC//AD$.  Denote as the figure attachment, then show that $KJLM$ be a parallelogram. S...

Let $A,B,C,D,E,F$ lie on a conic. Let $H=AC \cap BF$, $I = EC \cap BD$, $K=FD \cap AE$ $O=KI \cap CF$, $N=HI \cap AD$, $P=HK \cap BE$ , The...

Mở rộng định lý Napoleon kết hợp với một lục giác: Cho $ABCDEF$ là một lục giác bất kỳ, dựng ba tam giác đều $AGB$, $CHD$, $EIF$ cùng ...

1, Let $\lambda_1+\lambda_2+...+\lambda_n=1$ , $\lambda_i>0$   for  $ i=1,2,..,n$; f,f' are real positive continuous function that i...

Problem 1: Let $ABCD$ be a quadrilateral, $E$ be the midpoint of $BC$, $F$ be the midpoint of $AD$. The line though $E$ and perpendicular ...

Let $ABC$ be a triangle, let a line $L$ meets the Euler line at $D$ and the line $L$ meets three sidelines $BC,CA,AB$ at $A_0,B_0,C_0$ resp...

Let $ABC$ be a triangle, Let $(W)$ be the circle through $X(3),X(110)$ and $X(4240)$ in $ETC$. I found 7 points also lie on $(W)$ as follow...

Cho tam giác $ABC$ nội tiếp $(O)$. Một đường thẳng $d$ qua $O$ cắt $BC,CA,AB$ tại $A_0,B_0,C_0$. Gọi $A_b,A_c$ là hình chiếu của $A_0$ trên...

X(5463) = REFLECTION OF X(13) IN X(2) Each of the following sets of 4 points are concyclic: {X(13), X(15), X(5463), X(5464)}, {X(14), X(1...

Let $ABC$ be a triangle, let the Euler line of $ABC$ meets $BC,CA,AB$ at $A_1,B_1,C_1$ respectively. Define $A_2=X_{125}$ of the trian...

Lemma 1: Let a rectangular hyperbola with center $O$, let $P$ be a point on the plain such that the polar of $P$ to the hyperbola meets th...

Let ABC be a triangle, let P be a point on the line $X(13)X(15)$ (or $X(14)X(16)$). Three line through $P$ and perpendicular to $BC$ meets ...

Very nice theorem: http://tube.geogebra.org/student/m542855 Let ABC be a triangle, $F$ be the first (or secon) Fermat point, let $K$ be t...

Let a rectangular circumhyperbola of ABC, let L is the isogonal conjugate line of the rectangular hyperbola. The tangent line of the hyperb...

Let ABC be a triangle, let $A_0B_0C_0$ be Kiepert triangle of ABC. The circle with diameter $AA_0$ meets  $BC$ at $A_bA_c$. Define $B_a,B_c...

Problem:(Own)  Let ABC be a triangle, P be a point on the plaine, please show that three pair Thebault circle respecto AP,BP,CP alway lie o...

Please see: http://mathworld.wolfram.com/ChaslessPolarTriangleTheorem.html http://forumgeom.fau.edu/FG2004volume4/FG200427.pdf I propo...

Let $ABC$ be a triangle, let $P$ be the point $X(15)$ or $X(16)$. Let $A_0$ be a point on the plain such that: $\angle A0BP=\angle A0CP = ...

Theorem: Let $ABC$ be an equilateral triangle, let $P$ be a point on the plain, let $P_a,P_b,P_c$ be projection foot of $P$ on three sideli...

Let $ABC$ be a triangle $M_a$ is midpoint of $BC$, $H$ is the orthocenter of the triangle $ABC$. Let $V_A$, $V_B$, $V'_A$, $V'_B$ a...

 Let two conics through four common points $A,B,C,D$. Tangent line of the first conic  at $B,D$ meets the second conic at $G,H$ respectivel...

 I.In Forum Geometricorum Jounal Web site Forum Geometricorum 1- Đào Thanh Oai, Volum 14, Issue 10,   A simple proof of Gibert's g...

Theorem: Let three conics and one of them touch with fourth conic at two points. Then remaining pair of common tangents intersect at three...

Cite Course: Cut The Knot

Let ABC be a triangle, let $\triangle J_aJ_bJ_c$, $\triangle J'_aJ'bJ'_c$ be two Jacobi triangle, $A_0=J_aJ'_a \cap BC$, d...

Let $A,B,C,C_1,B_1A_1$ lies on a conic. Let $A_2=BC_1 \cap B_1C, B_2=AC_1 \cap A_1C; C_2=AB_1 \cap A_1B$. Let $A_3,B_3,C_3$ lie on sideline...

Problem 1: Let $ABC$ be a triangle, let a line $(L)$ through circumecenter and a point $P$ lie on circumcircle. Let $AP,BP,CP$ meets...

If ABCD and $A_1B_1C_1D_1$ are two similar rectangles (mean $AB:AD=A_1B_1:A_1D_1)$, then  Red=Yellow  <=>  $area(AA_1B_1B)+area(CC...

Colling theorem:   Let a line $(L)$ through the orthocener of the triangle ABC, reflection of $(L)$ in three sidelines concurrent be a poin...

X(4240) in Kimberling Center Click to downloand detail defined and the figure Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,...

Let a Tucker circle cut sidelines of given triangle $ABC$ at $A_b,A_c, B_c,B_a,C_a,C_b$. If $A_2A_bA_c, B_2B_aB_c, C_2C_aC_b$ are three sim...

Let $ABC$ be a triangle, $B_a,C_a$ lie on $BC; C_b,B_c$ lie on $CA,BA$ respectively. Such that two circles $(BB_aB_c),(CC_aC_b)$ tangent a...

Let $ABC$ be a triangle, let three points $A',B',C'$ on the circumcircle, such that $AA'//BB'//CC'$. Then the tria...

Let $ABC$ be a triangle, let three points $A_0,B_0,C_0$ lie on $BC,CA,AB$ such that $A_0,B_0,C_0$ are collinear. Let circle $(P)$ which ce...

- X(182) is midpoint of symmedian point X(6) and circumcenter X(3) - X(98) is intersection of Kiepert hyperbola and circumcircle Show ...

Let $ABC$ be a triangle, let a line $(d)$ cut the sidelines $BC,CA,AB$ at $A_1,B_1,C_1,$ denote $H_A,H_B,H_C$ are the orthocenter of $AB_1...

Let $ABC$ be a triangle, let a line $(d)$ cut the sidelines $BC,CA,AB$ at $A_1,B_1,C_1$, denote $H_A,H_B,H_C$ are the orthocenter of $AB_1...

Let a polynomial function:  f(x)=$x^n+a_{n−1}x^{n−1}+⋯+a_2x^2+a_1x+a_0$ Where n is positive integer, $n \geq 3$ and $a_0, a_1, a_2, ..., ...

A Generalization Lester theorem follow:  Theorem:  Let two point   lie on one branch of a rectangular hyperbola. The line pass through mi...

Giới thiệu:  Như ta đã biết luôn có một conic đi qua năm điểm phân biệt trên mặt phẳng. Nhưng với dấu hiệu nào ta có thể nhận ra đường con...