Lemma: Let (C) be a circle, X,X' two antipodal points on (C), and P a point. The centroid of the triangle PXX' is a fixed point. The loci of the H's, O's, N's...
Dear Antreas The locus of the incenter and the excenters looks like some high degree curve. As for the Lemoine point K, I think the locus is a circle. I name Y...
Dear Antreas! The locus of L is the circle and the locus of I the curve with degree 4. This can be obtained by simple coordinate calculation. It is interesting...
Alexey.A.Zaslavsky
zasl@...
Oct 1, 2008 9:10 am
16823
Dear Antreas ... I have corrected a typo in my previous message (I've forgotten a factor 2) PM^2+PN^2 and MN^2 are both constant (PM^2+PN^2 = 2(r^2+OP^2) where...
Looking at http://www.cut-the-knot.org/triangle/index.shtml I was asking myself how to find the vertices of ABC, with straightedge and compass, knowing the...
Dear Hyacinthists, Construct a triangle ABC given 1. (A,a,r_a+r) ok 2. (A,a,r_a-r) Datum ok 3. (A,a,r_b+r_c) Datum ok 4. (A,a,r_b-r_c) ok 5. (B,a,r_a+r) ? 6....
Dear Alexey, Francois, Jean-Pierre Thanks for your responses. In the problem: [APH] ... we got the centroid of PXX' fixed point. How about if, by construction,...
Old Theorem: Let A1B1C1, A2B2C2 be the cevian triangles of two points P1,P2. Denote: A* := P2A2 /\ B1C1 B* := P2B2 /\ C1A1 C* := P2C2 /\ A1B1 The triangles...
Hi everybody once again. 5.(B,a,r_a+r) ... Geometric construction: draw BC with length a and a line c through B forming an angle equal to B/2 with A. Take a...
Dear Eli! The triangle can't be constucted by the ruler and the compass if its three centers are given. Any construction problem which can be made by ruler and...
Alexey.A.Zaslavsky
zasl@...
Oct 2, 2008 7:29 am
16831
... Draw BC with length a and a line l_b through B forming an angle equal to B/2 with a. Choose a candidate b-excenter I_b on l_b and call I_c the intersection...
Dear Eli! The triangle can't be constucted by the ruler and the compass if its three centers are given. Any construction problem which can be made by ruler and...
Dear Friend, I think you can see some facts related this problem here: http://www.math.fau.edu/yiu/regularheptagontrisection070424.pdf Best regards, Bui Quang...
Dear Hyacinthists, Jack D'Aurizio, 5. (B,a,r_a+r)8. (B,a,r_b-r_c) Very nice constructions, thank you. Trying an algebraic solution to 5. I was expecting a...
... Dear Jean-Pierre, this is a quite astonishing lemma. Can you give me a proof (or a link to a proof) of this fact? Thank you in advance, Jack [Non-text...
I've corrected a typo and deleted my previous message. Re: [EMHL] NO Straightedge and compass construction of ABC from OHI, but... Dear Jack ... any ... This...
... - so, by knowing only OHI, it is possible to draw the Conway-conjugate of P wrt to ABC (the isogonal conj of the isotomic one) and to have an iterative ...
Let ABC be a triangle, and A'B'C' the pedal triangle of H (orthic triangle). Denote: Lab := The Euler Line of AA'B' Lac := The Euler Line of AA'C' A* := Lab /\...
Let ABC be a triangle, and A'B'C' the pedal triangle of O (medial triangle). Denote: Ab := OA' /\ CA Ac := OA' /\ AB Lab := The Euler Line of AbA'B' Lac := The...
... Only some preliminary considerations. Lac and Lab are parallel, being the Euler lines of two homothetic triangles B'A'Ab and C'AcA', and the same holds for...
... triangles ... translation bringing ... There are two more homothetic triangles with no common vertex. (in my figure, the triangles bounded by the lines...
... There isn't. They have parallel side of equal lengths, so they are corresponding in a translation (that's what I meant by "congruent"). Image attached. ...
Let A be a point, L a line, and (K) a circle. Let ABC be a triangle with vertex A the point A, vertices B,C lying on (K), and circumcenter O lying on L. Where...
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