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Topic: Three half-circles
Replies: 4   Last Post: Sep 6, 2011 8:51 PM

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Avni Pllana

Posts: 546
Registered: 12/6/04
Re: Three half-circles
Posted: Sep 6, 2011 3:06 PM
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> Let DEF be a rectangular triangle such that D=[0 0
> 0], E=[3 0 0] and F=[0 4 0]. On the sides DE, DF and
> EF are constructed respective half-circles with
> positive z-axis and perpendicular to the xy-plane.
> Determine the parameters A, B, C of the plane A*x +
> B*y + C*z = 1 , that tangents the three
> half-circles.
>
> Best regards,
> Avni



In the general case let EF = a , FD = b , DE = c , and

D=[0 0 0], E=[c 0 0], F=b*[ca , sqrt(1-ca^2) , 0] ,

where ca=(b^2+c^2-a^2)/(2*b*c) .

In this case the parameters A, B, C are as follows

A = (b^2-a^2)/(c*b^2) ,
B = (c^2*(a^2+b^2)-(a^4+b^4))/(4*b^2*c*S) ,
C = 2*a/(b*c) ,

where S is the area of triangle DEF.

Let D1, E1, F1 be the projections on the xy-plane of the respective touching points of the half-circles with the tangent plane. The lines DD1, EE1, FF1 are concurrent at the symmedian point X(6) of triangle DEF.
Furthermore, the intersection of the tangent plane with the xy-plane coincides with the tripolar line of the symmedian point.

Best regards,
Avni



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