porisms

1202

The blue triangles are equialteral. Prove $a^2+b^2=c^2$.

1201

The green and blue triangles are equilateral. Prove the red lines are parallel.

1200

The blue and green shapes are squares. Prove the red angle measures $45^\circ$.

1199

Prove the blue angle measures $45^\circ$.

1198

Prove the red lines are concurrent.

1197

The green lines are parallel. The blue lines are parallel. Prove the red line segments are congruent.

1196

The blue points are midpoints of sides of the yellow triangle. Prove the red line segments are congruent.

1195

Prove the red points are collinear.

1194

Prove the red lines are parallel.

1193

The yellow shape is an arbelos. Prove the red and blue points are collinear. Prove the red and green points are collinear.

1192

The black and green shapes are parallelograms. Prove the red lines are concurrent.

1191

Prove the red lines are concurrent if and only if $a^2+b^2+c^2=p^2+q^2+r^2$.

1190

Prove the red line segments are congruent.

1189

Prove $c=\sqrt{ab}$ and $b/c=\phi$.

1188

Prove the green triangle is isosceles right with area $a^2/2$.

1187

The black triangle is equilateral. Prove the red angle measures $60^\circ$.

1186

Prove the red and blue angles are congruent.

1185

Prove the red circles are congruent and their radius is the geometric mean of the radii of the blue circles.

1184

Prove the radius of the red circle is the geometric mean of the radii of the blue circles.

1183

Prove $ap+cr=bq+ds$.

1182

Prove the red lines are perpendicular if and only if the radius of the large circle equals the sum of the radii of the two small circles.

1181

The green line segments are congruent. The blue line segments are congruent. Prove the red line segment is the diameter of the circle.

1180

The green lines are parallel to the sides of the black triangle. Prove the sum of the radii of the yellow circles equals the radius of the blue circle.

1179

Prove the perimeter of the black triangle is $\dfrac{2a^2}{a-b}$.

1178

Find $x$.

1177

The gray lines are parallel to the sides of the black triangle. Prove the blue area equals the green area.

1176

Prove the blue and red angles are congruent.

1175

Prove the red lines are perpendicular.

1174

The blue triangles are similar with corresponding sides parallel. Prove the area of the red triangle is the geometric mean of the areas of the blue triangles.

1173

Prove $a:b=1:3$.

1172

Prove the red points are concyclic.

1171

Prove the red points are concyclic.

1170

Prove the red points are concyclic.

1169

Prove the green angles are congruent.

1168

Prove the red diagonals are perpendicular if and only if $a^2+c^2=b^2+d^2$.

1167

Prove the red points are concyclic.

1166

Prove $c=\sqrt{ab}$.

1165

Prove $x=\dfrac{a}{2a^2+1}$.

1164

Prove the area of the red rhombus is $\dfrac{8a^3b^3}{(a^2+b^2)^2}$.

1163

Prove the area of black trapezoid equals the area of the triangle with sides $a$, $b$, $2c$.

1162

The red angles are complements. Prove the black triangle is right.

1161

The green circles are congruent and concurrent. Prove $\dfrac1a+\dfrac1b=\dfrac1c$.

1160

The red line segments are congruent, concurrent, and parallel to the sides of the triangle. Prove $\dfrac1a+\dfrac1b+\dfrac1c=\dfrac2d$.

1159

Prove $\dfrac1a+\dfrac1b=\dfrac1c$.

1158

Prove the red line segment is the side of the inscribed decagon.

1157

Prove the red line segments are congruent.

1156

Prove the red, blue, and green points are concyclic.

1155

Prove $\dfrac1a+\dfrac1b=\dfrac2c$.

1154

Prove $\text{green area}:\text{red area} = 1:3$.

1153

Prove $\text{yellow area}:\text{red area} = 3:16$.

1152

The green and blue points are the centroids of the green and blue triangles. Prove the red line segments are congruent.

1151

Prove ratio of the sides of the triangle is $3:3:4$.

1150

The black shape is a parallelogram. Prove the blue area equals the red area.

1149

Prove ratio of the sides of the triangle is $5:10:13$.

1148

Given a parallelogram. Prove the red line segment is the radius of the red circle.

1147

Prove the red circle is tangent to a leg of the trapezoid if and only if the green circle is also.

1146

The green points are the midpoints of the sides of the triangle. The blue points are the midpoints of the altitudes of the triangle. Prove the red lines are concurrent.