Let H be the orthocenter of acute △ A B C . . Points D and M are defined as the projection of A onto segment BC and the midpoint of segment BC, respectively. Let H ′ be the reflection of orthocenter H over the midpoint of DM, and construct a circle Γ centered at H ′ passing through B and C. Given that Γ intersects lines AB and AC at points X ≠ B and Y ≠ C respectively, show that points X, D, Y lie on a line.

Question

Let H be the orthocenter of acute   △  A  B  C  . . Points D and M are defined as the projection of A onto segment BC and the midpoint of segment BC, respectively. Let       H    ′   be the reflection of orthocenter H over the midpoint of DM, and construct a circle   Γ centered at       H    ′   passing through B and C. Given that   Γ intersects lines AB and AC at points   X  ≠  B and   Y  ≠  C respectively, show that points X, D, Y lie on a line.

pheniankang · Accepted Answer

Step 1Hints:  ∠  B  X  Y  =  ∠  B  C  YSo:  △  P  Y  C  ∼  △  P  B  XSo we have:            B      P              P      Y        =            X      P              P      C      Or:  B  P  ×  P  C  =  X  P  ×  P  YNow draw circle L centered on midpoint of BC, so it passes B and C(BC is it&#039;s diameter). It intersect Altitude AD at Q. Draw another circle S diameter as XY(centered on midpoint of XY).Draw third circle T centered on D with radius QD, it intersect circle S at point R. In right angle triangle QBC we have:  Q      D    2    =  B  D  ×  D  CIn right angled triangle XRY we have:  X  D  ×  D  Y  =  R      D    2  So we have:  R      D    2    =  Q      D    2    ⇒  B  D  ×  D  C  =  X  D  ×  D  YThat means   △  B  D  X  ∼  △  D  Y  CSince YC and BX are also sides of triangle YPC and BPX, this give the result that P is coincident on D or point X, D and Y are on one line.

Let H be the orthocenter of acute <mi mathvariant="normal">△ A B C

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