If a triangle is inscribed in a n ellipse and two of its sides are parallel to the given straight lines, then prove that the third side touches the fixed ellipse.

An equilateral triangle is inscribed in an ellipse whose equation is x^2+4y^2=4 If one vertex of the triangle is (0,1) then the length of each side is

A quadrilateral is inscribed in a parabola y^2=4a x and three of its sides pass through fixed points on the axis. Show that the fourth side also passes through a fixed point on the axis of the parabola.

Consider the ellipse whose major and minor axes are x-axis and y-axis, respectively. If phi is the angle between the CP and the normal at point P on the ellipse, and the greatest value tan phi is 3/4 (where C is the centre of the ellipse). Also semi-major axis is 10 units . A rectangle is inscribed in a ellipse whose sides are parallel to the coordinate axes, then the maximum area of rectangle is

The ellipse E_(1) : x^(2)/9 + y^(2)/4 =1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E_(2) passing through the point (0,4) circumscribes the rectangle R. The eccentricity of the ellispe is

Prove that line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

A triangle has sides 6,7, and 8. The line through its incenter parallel to the shortest side is drawn to meet the other two sides at P and Q. Then find the length of the segment PQ.

An ellipse slides between two perpendicular straight lines. Then identify the locus of its center.

If two side of a triangle are hat i+2 hat ja n d hat i+ hat k , then find the length of the third side.

An equilateral triangle is inscribed in the parabola y^(2) = 4ax whose vertex is at the vertex of the parabola. Find the length of its side.