Dear mathematician friends,
I have a question for you, if I may. My children and I have been playing with classic geometry problems. We’ve come across this technique of constructing a tangent to a point on a circle. It involves drawing a series of circles at intersecting points.
I have a question for you, if I may. My children and I have been playing with classic geometry problems. We’ve come across this technique of constructing a tangent to a point on a circle. It involves drawing a series of circles at intersecting points.
While I find it perfectly fun to draw beautiful circles to make a tangent, it has been gnawing on me for a whole day and night that I cannot seem to figure out why it results in a line that’s at the required right angle to the radius. Argh! This is what I’ve got.
No matter how much I move the angles around in various triangle-angle-sum equations, I don’t get anywhere illuminating.
Can anyone explain this to me? Please?
Can anyone explain this to me? Please?
I should clarify, the tangent is the result, not the beginning. This is how the problem starts: you have a circle, its centre, and a point on the circle. You have a compass and a ruler. How to construct a tangent in only three steps?
Step 1: draw a circle with the radius going from another point on the circle to the given point on the circle.
Step 2: draw another circle, this time from the given point on the circle to the intersecting point of the two circles from step 1.
Step 3: draw a line between the given point on the circle and the new intersecting point between the circles from step 1 and 2. Done.
Mystery!
Or, hair-pulling puzzlement, as the case may be.
Mystery!
Or, hair-pulling puzzlement, as the case may be.
Got it!
My hurdle: I was missing the inscribed angle theorem. Tomáš pointed to it with elegant efficiency.
The cyan angle referred to I labeled ε though it is hard to see; I was trained in the Greek letter notation for angles, and I still like it best.
math.stackexchange.com/questions/1961…
My hurdle: I was missing the inscribed angle theorem. Tomáš pointed to it with elegant efficiency.
The cyan angle referred to I labeled ε though it is hard to see; I was trained in the Greek letter notation for angles, and I still like it best.
math.stackexchange.com/questions/1961…
Thanks to everyone who joined in!
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