Prove that circle with diameter $BN$, circle with diameter $CM$ and Euler circle of traingle $ABC$ concur.
$begingroup$
$ABC$ is an acute triangle which is inscribed circle $(O)$. A line through $O$ cut $AB,AC$ at $M,N$. Prove that circle with diameter $BN$, circle with diameter $CM$ and Euler circle of traingle $ABC$ concur.
I don't know the way to solve this. I think it is impossible to use basic Euclid geometry theorem. There are lots of points on circles, where we can use pascal theorem, but how to use? Please solve me the added geometry factor to draw. Thanks.
geometry
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
$endgroup$
add a comment |
$begingroup$
$ABC$ is an acute triangle which is inscribed circle $(O)$. A line through $O$ cut $AB,AC$ at $M,N$. Prove that circle with diameter $BN$, circle with diameter $CM$ and Euler circle of traingle $ABC$ concur.
I don't know the way to solve this. I think it is impossible to use basic Euclid geometry theorem. There are lots of points on circles, where we can use pascal theorem, but how to use? Please solve me the added geometry factor to draw. Thanks.
geometry
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
$endgroup$
$begingroup$
There is a simple approach: apply a circle inversion with respect to the circle with diameter $BC$.
$endgroup$
– Jack D'Aurizio
Dec 19 '18 at 20:59
add a comment |
$begingroup$
$ABC$ is an acute triangle which is inscribed circle $(O)$. A line through $O$ cut $AB,AC$ at $M,N$. Prove that circle with diameter $BN$, circle with diameter $CM$ and Euler circle of traingle $ABC$ concur.
I don't know the way to solve this. I think it is impossible to use basic Euclid geometry theorem. There are lots of points on circles, where we can use pascal theorem, but how to use? Please solve me the added geometry factor to draw. Thanks.
geometry
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
$endgroup$
$ABC$ is an acute triangle which is inscribed circle $(O)$. A line through $O$ cut $AB,AC$ at $M,N$. Prove that circle with diameter $BN$, circle with diameter $CM$ and Euler circle of traingle $ABC$ concur.
I don't know the way to solve this. I think it is impossible to use basic Euclid geometry theorem. There are lots of points on circles, where we can use pascal theorem, but how to use? Please solve me the added geometry factor to draw. Thanks.
geometry
geometry
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
asked Dec 19 '18 at 14:51
Trong TuanTrong Tuan
1318
1318
$begingroup$
There is a simple approach: apply a circle inversion with respect to the circle with diameter $BC$.
$endgroup$
– Jack D'Aurizio
Dec 19 '18 at 20:59
add a comment |
$begingroup$
There is a simple approach: apply a circle inversion with respect to the circle with diameter $BC$.
$endgroup$
– Jack D'Aurizio
Dec 19 '18 at 20:59
$begingroup$
There is a simple approach: apply a circle inversion with respect to the circle with diameter $BC$.
$endgroup$
– Jack D'Aurizio
Dec 19 '18 at 20:59
$begingroup$
There is a simple approach: apply a circle inversion with respect to the circle with diameter $BC$.
$endgroup$
– Jack D'Aurizio
Dec 19 '18 at 20:59
add a comment |
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$begingroup$
There is a simple approach: apply a circle inversion with respect to the circle with diameter $BC$.
$endgroup$
– Jack D'Aurizio
Dec 19 '18 at 20:59