08 Circles with diameters equal to corresponding sides of the triangle

From the figure shown below, O1, O2, and O3 are centers of circles located at the midpoints of the sides of the triangle ABC. The sides of ABC are diameters of the respective circles. Prove that

$A_1 + A_2 + A_3 = A_4$


where A1, A2, A3, and A4 are areas in shaded regions.

Circles with centers at midpoints of sides of a right triangle


Conic Sections

Conic sections can be defined as the locus of point that moves so that the ratio of its distance from a fixed point called the focus to its distance from a fixed line called the directrix is constant. The constant ratio is called the eccentricity of the conic.

01 Arcs of quarter circles

Example 01
The figure shown below are circular arcs with center at each corner of the square and radius equal to the side of the square. It is desired to find the area enclosed by these arcs. Determine the area of the shaded region.

Intersection of circular quadrants


01 Rectangle of maximum perimeter inscribed in a circle

Problem 01
Find the shape of the rectangle of maximum perimeter inscribed in a circle.

The Circle

The following are short descriptions of the circle shown below.

Tangent - is a line that would pass through one point on the circle.
Secant - is a line that would pass through two points on the circle.
Chord - is a secant that would terminate on the circle itself.
Diameter, d - is a chord that passes through the center of the circle.
Radius, r - is one-half of the diameter.



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