1-Let P is radical center of three circles (A),(B),( C ). Let circle (P).
- (a) is radical axis (P) and (A)
- (b) is radical axis (P) and (B)
- ( c ) is radical axis (P) and (C)
+ (a) intersect BC at A1
+ (b) intersect AC at B1
+ ( c ) intersect AB at C1
Prove that: A1,B1,C1 are collinear
Prove that: A1,B1,C1 are collinear
2-From a generalization of Simson line to a generalization Droz-Farny line.
Let a triangle ABC and D points. DA,DB,DC intersect the (ABC) circle at A1,B1,C1. Let N is a point on the circle. NA1,NB1,NC1 intersect BC,CA,AB at A2,B2,C2. We have A2,B2,C2 are collinear(The line A2B2 are generalization double Simson Line) reference: http://www.cut-the-k...scalConic.shtml. A3,B3,C3 are circum of three circle (DA1A2) , (DB1B2),(DC1C2) . Prove that A3,B3,C3 are collinear(A3B3 are a generalization Droz-Frany line) and (DA1A2) , (DB1B2),(DC1C2) concurrent at secon point are on (ABC). N is a fixed point, when N move on (ABC) then A3B3 tangent with a conic.
I discovered Four new line on 17-September-2013
3-Let a triangle ABC. Let A1,B1,C1 are on BC,CA,AB and A1,B1,C1 are collinear. Construct three parallel line through A1,B1,C1. The line through A1 intersect AB,AC at A1c,A1b(respectively). The line through B1 intersect BA,BC at B1c,B1a(respectively). The line through C1 intersect CA,CB at C1b,C1a(respectively). A2,B2,C2 are midpoints of A1cA1b; B1aB1c; C1aC1b. We have A2,B2,C2 are collinear.
4-A new property of Fermat points
Let a triangle ABC and F is Fermat point of the triangle ABC. N,M,P are on BC,BA,AC and angleNFM=60deg; angleMFP=60 deg; angleNFP=60 deg(Show in the figure attachment). Prove that N,M,P are collinear.
5 and 6. Let a triangle ABC and A1,B1,C1 are on BC,CA,AB and A1,B1,C1 are collinear. A2,B2,C2 are midpoint of B1C1,C1A1,A1B1(respectively). AA2,BB2,CC2 interect BC,CA,AB at A3,B3,C3 respectively. A4,B4,C4 are midpoints of AA3,BB3,CC3 respectively. Prove that:
5- A3,B3,C3 are collinear
6-A4,B4,C4 are collinear
Bài viết đã được chỉnh sửa nội dung bởi khongghen: 19-09-2013 - 09:04
