2199

Prove the sum of red line segments is $$\sqrt{\frac{1}{2}\left(a^2+b^2+c^2\right) + 2A\sqrt{3}}$$ where $A$ is the area of yellow triangle.

2198

Prove the red line is parallel to sides of the blue parallelogram.

2197

The red lines trisect the sides of the yellow quadrilateral. Prove the red line segments trisect each other.

2196

Prove $c=\frac{1}{2}(a+b)$.

2195

Prove the red medians trisect the blue line segment.

2194

Prove the red points are concyclic.

2193

Prove the red line segments are congruent.

2192

Prove the radius of the red circle equals the semiperimeter of the right triangle.

2191

The blue lines are parallel to the sides of the triangle. Prove $ABC=8DEF$.

2190

Prove the red angles are congruent.

2189

Prove $c=\frac{b}{a}(a+b)$.

2188

Find the measures of the red angles.

2187

Prove the red angles are congruent.

2186

Prove the red line is tangent to the green circle.

2185

Prove the red and purple points are collinear iff the blue and purple points are collinear.

2184

Prove the red angles are congruent.

2183

The yellow shape is a square. Prove the red angle is double the blue angle.

2182

The blue angle is double the green angle. The blue line segment is double the green line segment. Prove the red line bisects a side of the yellow triangle.

2181

The blue line bisects the legs of the right triangle. Prove the red points are concyclic.

2180

Prove the blue angles are congruent iff the red angles are congruent.

2179

Prove the red point trisects the blue line segment.

2178

The green circles are tangent to the yellow circle. Prove the red line bisects the blue side of the triangle.

2177

The yellow shape is a parallelogram. Prove the red lines are perpendicular.

2176

The yellow shape is a trapezoid. Prove the red angles are congruent.

2175

The blue shapes are squares. The yellow triangle is equilateral. Prove the red points are collinear.

2174

Find the measure of the red angle.

2173

The blue points are the midpoints of the sides of the green triangle. Prove the perimeter of the red polygon equals the perimeter of the green triangle.

2172

The yellow and blue shapes are squares. Prove the red and green points are collinear and the green point is the midpoint of the red points.

2171

The yellow shape is a parallelogram. The green line bisects the green angle. The blue line bisects the blue angle. The red line bisects the red angle. Prove the red and yellow lines are perpendicular.

2170

Prove the area of the triangle is $\frac{1}{2}r_c(a+b-c)$ (cf. $\frac{1}{2}r(a+b+c)$).

2169

Prove the red line segments are congruent.

2168

A circle and square intersect at eight points. Prove the sum of the red arcs equals the sum of the blue arcs and the sum of the red perimeters equals the sum of the blue perimeters.

2167

Prove red line bisects the blue side of the triangle.

2166

Prove the blue and red angles are supplementary and $a:b = c^2:d^2$.

2165

Prove $r=\frac{1}{2}\sqrt{m^2+n^2}$ .

2164

The yellow shape is a parallelogram. Prove the red lines are perpendicular.

2163

Prove the area of the trapezoid is $\sqrt{mn}\left(b+m+\dfrac{mn}{b-n}\right)$.

2162

The blue angles differ by $60^\circ$. The green angles differ by $60^\circ$. The yellow angles differ by $60^\circ$. Prove the red triangle is equilateral.

2161

The blue circles are congruent. Prove the red lines are parallel.

2160

Find the measure of the red angle.

2159

Prove the red lines are perpendicular.

2158

Prove the red line segments are congruent.

2157

Prove the blue base angle is double its opposite angle iff the red base angle is double its opposite angle.

2156

Prove the circles are tangent to each other.

2155

Prove the red and blue areas are equal.

2154

The blue hexagon is equiangular. Prove the red and green areas are equal.

2153

Given a regular dodecgaon. Prove the red line segments are congruent.

2152

Prove the red point trisects the blue line segment.

2151

The green point is the midpoint of the yellow arc. Prove the red line segment is the sum of the circumradii of the blue circles.

2150

Prove the circles are concurrent.

2149

Prove the red points are concyclic.

2148

Prove the area of the regular dodecagon is $d^2$.

2147

Prove the red angle is $45^\circ$.

2146

Prove the red line bisects the blue arcs.

2145

Prove the red and blue areas are equal.

2144

The green and yellow rectangles are similar. The sides of the yellow rectangle are parallel to the sides of the green rectangle. Prove the red and blue lines are concurrent.

2143

The blue lines are tangent to the circle. Prove the red lines are also.

2142

Prove the red and blue lines are concurrent and the blue lines bisect the red line segment.

2141

Prove the area of the blue quadrilateral is double the area of the red triangle.

2140

Prove the red points are concyclic.