Squares and Triangles in a Circle

Square ABCD is inscribed in circle O with radius 30. Square EFGH is in a minor circular segment. E and F are on arc AB, while G and H are on cord AB. What is the length of EF?

Equilateral triangle ABC is inscribed in circle O with radius 30. Square EFGH is in a minor circular segment. E and F are on arc AB, while G and H are on cord AB. What is the length of EF?

Equilateral triangle ABC is inscribed in circle O with radius 30. Equilateral triangle EFG is in a minor circular segment. E and F are on arc AB, while G is on cord AB. EF//AB. What is the length of EF?

Triangle inside a Triangle

In triangle ABC, D, E, and F are on AB, AC, and BC respectively. AD:BD = 2:1; AE:EC = 1:1; BF:FC = 3:1.  AF and BE intersects at G; BE and CD intersects at H; AF and CD intersects at I. If the area of triangle ABC is 504, what is the area of triangle GHI?

Circle in the Corner

A regular hexagon ABCDEF has the side length of 6. Circle O is inscribed in the hexagon. The tangent points on AB and AF are G and H respectively. Circle P is inscribed in the corner area sounded by HA, AG, and arc GH. What's the radius of circle P? (Do not use calculator. Let the square root of 3 be 1.73. When submitting the result, keep two digits after decimal point.)

Circle in a Right Triangle

A circle is inscribed in a right triangle of leg lengths 8 and 15. What's the area of this circle? (Let Pi = 3.14.)

Dissecting a Triangle

ABC is a triangle. D and E are on AB and AC respectively. BE and CD intersects at F. The areas of triangles CEF, BDF, and BCF are 10, 15, and 20 respectively.  What is the area of quadrilateral AEFD?

Angle LEO

A regular 26-sided polygon has its 26 vertices marked as A, B, C, .... , Z clockwise. What's the angle LEO? (Please keep one digit after decimal point.)

Angles of a Star

ABCDE is a convex pentagon. What is the sum of the following five angles?

Regular Hexagon

ABCDEF is a regular hexagon. G, H, I, J, K, and L are midpoints on AB, BC, CD, DE, EF, and FA respectively. M is a point inside ABCDEF. Connecting M to the 6 midpoints dissects the hexagon into 6 small quadrilaterals. The vertices of each small quadrilateral includes a shared vertex with ABCDEF, two midpoints adjacent to the shared vertex, and the interior point M. The areas of the AGML, DJMI, and EKMJ are 41, 53, and 67, respectively. What’s the area of BHMG?

Hexagon

ABCDEF is a hexagon. G, H, I, J, K, and L are midpoints on AB, BC, CD, DE, EF, and FA respectively. M is a point inside ABCDEF. Connecting M to the 6 midpoints dissects the hexagon into 6 small quadrilaterals. The vertices of each small quadrilateral includes a shared vertex with ABCDEF, two midpoints adjacent to the shared vertex, and the interior point M. Starting from the quadrilateral with vertex A, going clockwise, the areas of the first five quadrilateral are 81, 64, 49, 36, and 25, respectively. What’s the area of the remaining quadrilateral (the one with vertex F)?