Same angle, same circle

The idea

If a line segment AB subtends equal angles at two other points C and D on the same side of AB, then A, B, C and D lie on a circle.

A hunt for D

You know that every point of a circle, on the far side of the arc, sees AB at the same angle. Turn the question round. A, B and C are fixed, and ∠ACB = 50°. You look for another point D, on the same side of AB as C, that also sees AB at 50°.

Where can D be, if ∠ADB = ∠ACB and D is on the same side of AB as C?

• Anywhere on that side of AB
• Only on a circle through A, B and C
• Only on a straight line

Hunt for D

A, B and C are fixed and the circle is hidden. Touch above AB to place D. When ∠ADB comes to equal ∠ACB, D locks on.

Only on the circle

If a line segment AB subtends equal angles at two other points C and D on the same side of AB, then A, B, C and D lie on a circle.

Draw the circle through A, B and C. If D were inside it, ∠ADB would be bigger than ∠ACB. If D were outside, it would be smaller. Since the two angles are equal, D has to be on the circle.

Notes

If AB subtends equal angles at C and D on the same side of AB, then A, B, C, D are concyclic.

Check yourself

C and D are on the same side of AB. ∠ACB = 40° and ∠ADB = 40°. What can we say about A, B, C and D?

C and D are on the same side of AB. ∠ACB = 40° and ∠ADB = 55°. Where is D, compared with the circle through A, B and C?

A, B, C and D lie on one circle, and C and D are on the same side of AB. ∠ACB = 36°. What is ∠ADB, in degrees?

Answer: 36 °

C and D are on the circle, on the same side of AB, so ∠ADB = ∠ACB = 36°.

∠ACB = 70° and ∠ADB = 70°, but C and D are on opposite sides of AB. Must A, B, C and D lie on one circle?

• They lie on one straight line. C and D are not on the line AB. Equal angles on one side mean the four points lie on a circle.
• AB is a diameter. We do not know that. Equal angles on one side only tell us that the four points lie on a circle.
• They lie on one circle — correct. Yes. Equal angles at C and D, on the same side of AB, mean the four points are concyclic.
• Inside the circle — correct. Yes. Inside the circle the angle is bigger, so 55° means D is inside.
• Outside the circle. Outside the circle the angle is smaller than 40°. Here it is bigger.
• On the circle. On the circle the angle would be exactly 40°.
• Yes, equal angles always give a circle. The rule needs C and D on the same side of AB. On opposite sides, equal angles do not give one circle.
• Not necessarily — correct. Right. The rule needs C and D on the same side of AB.
• Yes, because 70° + 70° = 140°. Adding the angles does not help. The rule needs C and D on the same side of AB.