UnivHypGeom2: Apollonius and harmonic conjugates

Apollonius introduced the important idea of harmonic conjugates, concerning four points on a line. He showed that the pole polar duality associated with a circle produces a family of such harmonic ranges, one for every line through the pole of a line. Harmonic ranges also occur in the context of vertex bisectors, as combinations of vectors, and associated with the sides of a quadrangle.

Category:

Education

Tags:

• UnivHypGeom2
• Hyperbolic
• geometry
• Wildberger
• Apollonius
• harmonic
• conjugates
• range
• projective
• quadrangle
• invariance
• cross
• ratio
• bisectors
• vectors
• circle
• polarity
• pole
• polar

[footstep002]

I've tried several times to solve exercise2.1 using CaR program. But I can't plot the tangent line for gamma. As you know it is easy to find a null point starting 'from a pole outside circle'. It's just a result of first definition of pole, that includes a tangent line.

But in exercise2.1 null point is specified first, and I'm keep failing to get a proper tangent line for that null point.

Could you tell me how can I solve this problem?

[njwildberger]

@footstep002 Take any line through the null point, and find its pole. Then use the fact that if line A passes through point b, then the dual point a lies on the dual line B.

[alexzarhin]

Hello,

After several unsuccessful attempts to solve the stared question "how to find a center of a circle using only a straightedge", I looked in the internet and found out that it is impossible, according Poncelet-Steiner theorem.

Is this what you ment or I misunderstood something?

With Respect,

Alex.

[njwildberger]

@alexzarhin This is what I meant. It is useful in universal hyperbolic geometry to realize that the usual `center' of the null circle is not a valid concept in projective geometry.

[QuantumMaths]

For exercise 2.1, could one simply draw a chord through the desired point of tangency, select two points on the chord, find the polars of both of them, intersect them, take that point, and then draw a line through the desired point of tangency and the intersection of the two polars? Two points define a line, right? So the third point generated by two other polars would be redundant... right?

[Tr3xKuro]

@alexzarhin

LOL NOW I KNOW WHY HE HAS THAT SMUG GRIN ON HIS FACE WHILE HE'S TELLING US TO ATTEMPT IT.

[sstl517]

@njwildberger , if you are willing to forgo your requirement of "using a strait-edge only" and use a compass also, here is a good illustration of the technique:

wwwmakeitsolardotcom/science-f­air-ideas/90-find-circle-cente­r.htm



[brangelito]

Thank you.

[danodet]

I love the way hierarchy of geometry is presented with respect to the toolkit allowed. I have a question about the carpenter’s square construction. Is it I or II ?

I. Given a line and a point on the line you can draw a line perpendicular to the given line that pass through the given point.

II. Given a line and a point in the plane you can draw a line perpendicular to the given line that pass through the given point.

It seems to me that II is stronger than I.

Thank you

[EclecticSceptic]

@EclecticSceptic Sorry could you ignore this question please, it's invalid for many reasons.