Geometrical Interpreation

[stuart clark]

Given 6 numbers which satisfy the relations:

y² + yz + z² = a²
z² + zx + x² = b²
x² + xy + y² = c²

Determine the sum x+y+z in terms of a, b, c.

Give geometrical interpretation if the numbers are all positive.

[Lokman Gökçe]

Let be ABC a triangle such that BC = a, CA = b, AB = c. Take a point P such that PA = x, PB = y, PC = z and m(APB) = m(BPC) = m(CPA) = 120^o. (actually, point P is First Fermat Point of triangle ABC).

Let's apply cosine theorem in the triangles APB, BPC, CPA. We yields given equations.

Calculation of x + y + z

x + y + z is minimum sum that from any point on the plane of triangle ABC to the vertex. Let's contruct (externally) a equilateral triangle BCD on side BC. x + y + z = AD. So our problem to reduce calculating of AD.

NOTE: thanks for your problems. I hope that you can calculate AD. But, if you want, I can write AD in terms of a, b, c. have a good days.

[stuart clark]

Thanks Admin.

would you like to explain me the solution after the line..

Calculation of x + y + z

x + y + z is minimum sum that from any point on the plane of triangle ABC to the vertex.

Let's contruct (externally) a equilateral triangle BCD on side BC. x + y + z = AD. So our problem to reduce calculating of AD.

Thanks

[Lokman Gökçe]

You may examine the following document about Fermat point.

[Lokman Gökçe]

I write a numerical problem and I give two solutions in the following document.

[stuart clark]

Thanks Admin.

[alpercay]

Aşağıdaki bağlantıda da tartışılmış. Burada bulunsun.

https://matkafasi.com/109559/x-0-olmak-uzere-x-xy-25-z-zx-64-ve-y-yz-49-ise-x-z-degeri