Points of Concurrency

[erfarmer]

Hello,

I am trying to figure out how (and if) points of concurrency are part of High School Geometry. I know that constructions are (G.CO.12 & G.CO.13), and there is a nod to the idea in G.CO.10 “Prove theorems about triangle. Theorems include: . . . the medians of a triangle meet at a point“. In Wu’s writing on Geometry – he explains how to prove each of the points are concurrent. So my question is to what extent should we focus on these points – their names? their properties? how to construct them? Any guidance or suggestions is appreciated.

[Kristie Donavan]

Does anyone have a response to this post? Our teachers are also asking about the extent of studying points of concurrency, if at all.
In reading the standards, I don’t interpret “points of concurrency” as a topic in itself; I see the concepts instead used in solving problems like in standards G-C.3 (construct inscribed and circumscribed circles of a triangle), G-CO.10 (prove medians of a triangle meet at a point), and G-CO.9 (points on perpendicular bisector are equidistant from endpoints). Am I interpreting this correctly?

[Bill McCallum]

First let me say that having grown up with a fairly traditional education in Euclidean Geometry in Australia I have never heard of “points of concurrency” as a topic. So I agree with Kristie Donavan!

I’m assuming this refers to the various theorems about medians, altitudes, angle bisectors, and side bisectors of triangles all intersecting at a point. The only one of these that is explicitly called out in the standards the one about medians. Constructing inscribed and circumscribed circles suggests also studying the concurrency of angle and side bisectors, although I think there is latitude in curriculum about how far you go with that. I myself would not advocate remembering all the names of the points where various lines intersect, and that is certainly not required by the standards, although of course it is not forbidden either.

Generally speaking the high school standards were designed to allow states some latitude in curriculum.

[Cathy Kessel]

Re “to what extent should we focus on these points – their names?”: Note that
the few standards that prescribe terms do so explicitly, e.g., 1.G.3: “describe the shares using the words halves, fourths, and quarters.”

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