1) The circles c

2) The products AH*HE = BH*HF = CH*HD.

3) H is the radical center of any three circles passing through points (A,E), (B,F), (C,D) respectively.

4) angle(D'CC') = |angle(B)-angle(C)|.

5) angle(AJE) = 2angle(B). Triangle AJE is isosceles.

6) Let L be the intersection point of c

7) Points L, H, J, C' are on a line, which bisects angle(ALE).

8) The symmetric P of D' w.r. to O, A, K and the intersection M of circumcircle of LAJE with BC are on a line.

9) CABP is an isosceles trapezium.

10) The Simson line P'P'' of P is parallel to CC'.

11) The orthocenter H and the circumcenter O are isogonal conjugate.

12) Triangle IKO is isosceles and angle(I)=angle(O) = angle(B).

13) The angle of lines e and KM is equal to angle(KIH).

Exercise: express the angle HC'D' in terms of triangle's data.

1) Follows by measuring the angles of triangle IJE.

2) Follows from the cyclicity of quadrilaterals FABE, BDFC, etc.

3) Follows from 2.

4) Follows by measuring the angles of triangle CDC'.

5) AJE is central to B. Note that IJ is parallel and half of CC' and apply 4.

6) Compare the opposite angles at J and L.

7) Follows from 6.

8) Measure angles at M.

9) Obvious.

10) Measure the angles at P' and B using 9.

11) Follows from the equality of angles angle(ACD)=angle(C'CB).

12) IO is parallel and half of HC'. The medial line of LJ must pass through the middle of IO. Besides LM, KO are both orthogonal to LA, hence parallel and angle(KOI) = angle(JLM) = angle(B).

Autopolar.html

Bisector.html

CircleBundleLocus.html

Droz_Farny.html

EulerCircle.html

EulerCircleProperty.html

GaussBodenmiller.html

Isogonal_Conjugation.html

MedialParabola.html

Miquel_Point.html

Orthocenter.html

OrthoRectangular.html

Polar4.html

RectHypeCircumscribed.html

RectHypeImpossible.html

RectHypeThroughFourPts.html

Simson.html

SimsonDiametral.html

SteinerLine.html

TrianglesCircumscribingParabolas.html

WallaceSimson.html

Produced with EucliDraw© |