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How do you find the circumcenter of a triangle on an isosceles triangle?

How do you find the circumcenter of a triangle on an isosceles triangle? I know you have to find the equation for two perpendicular lines to any two lines to find the circumcenter for any triangle, but with the isosceles triangle, there's a shortcut or even a different way. Does anyone know?

Public Comments

1. It depends on what information is given about the triangle. Do you have the coordinates of the vertices, or the lengths of the sides or...? One of the lines can be the line of symmetry. Here's a method: Place the triangle with the base on the x-axis and the line of symmetry on the y-axis. Make the equation of an altitude from on of the vertices on the x-axis to the opposite side. Set x=0 for that equation. Solve for y. That is the circumcenter. EDIT: I defer to starwhitedwarf. The circumcenter is at the intersection of the perpendicular bisectors of the sides, not at the intersection of the altitudes, called the orthocenter. EDIT2: Check out the link below for all kinds of centers in triangles. Bisecting all the angles gives the center of an inscribed circle or "incircle."
2. I have attempted to work out the co-ordinates of the circumcenter for an isosceles triangle with vertices at A(a, 0), C(-a,0) and B(0,b) The circumcenter will clearly lie on the y-axis. If you workout the equation of the perpendicular bisector of BC and see where it intersects the y-axis you get: (0, (b² - a²)/2b ) Any triangle in the plane would be a translation of the triangle and/or a rotation. Hence we can work out the co-ordinates of the cicumcenter for any triangle. Ofcourse, it is easiest when the triangle is simply a translation By say the vector (m,n) Then the circumcenter would be at (m, n + (b² - a²)/2b) I will leave you to work out the most general case, which is when you have the added effect of a rotation.
3. I don't remember it properly but out of these two any one may work. 1) Bisect all the angels , then by taking the distance between incident point of the three angle bisectors and any one vertex as radius, draw a circle. (circle is circumcircle and the centre as circumcenter.) ------------------------- OR ---------------------------- 2) Bisect all the sides , then by taking the distance between incident point of the three side bisectors and any one vertex as radius, draw the circle. (circle is circumcircle and the centre as circumcenter.) Lets' see if it works......
4. r u talking about a drawn isosceles triangle or just an isosceles triangle in general? i dont know how to find it in general but to get it from a drawn triangle u bisect all angles and the point at which they intersect is the circumcenter.