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Congruent Triangles 4.52 Importance of concurrency

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**Triangle Points of Concurrency**

#1 Perpendicular bisectors and the circumcenter.

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Circumcenter Before we can talk about the circumcenter’s importance, we need some review on perpendicular bisectors.

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**perpendicular bisector is equidistant from A and B.**

Review of Perpendicular Bisector Properties. Every point on the perpendicular bisector is equidistant from A and B.

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**Therefore, the circumcenter is equidistant from**

each vertex. Why ?

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**Since C is on each perpendicular bisector it is equidistant from each segment’s endpoint.**

Therefore…

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**Ergo the term… Circumcenter.**

The equal distances are radii for a circle that is … written around (circumscribed) the circle. Ergo the term… Circumcenter.

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Why is this important? If A, B, and T are three cities, Point C is the ideal place to build a communication tower to broadcast to each city.

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**Triangle Points of Concurrency**

#2 Angle bisectors and the Incenter.

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**From any general point on the angle bisector, the perpendicular distance to either side is the same.**

Why ? Y X P

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**The top angles are congruent by definition of angle bisector.**

There are two right angles are congruent by def. of perpendicular. Y by the reflexive property of = X ? By AAS P ? By CPCTC

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**Let’s see. Why is this important ? Each new point creates**

2 congruent triangles by AAS. Why is this important ? Let’s see.

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**Since the incenter P is on each angle bisector….**

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**P Why is this important? Point P is equidistant from each side.**

Remember, the distance from a point to a line is the brown perpendicular segment. P Why is this important? Because these equal distances are radial distances.

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**Radial distances refer to a circle.**

The incenter generates an inscribed circle. P

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**Triangle Points of Concurrency**

#3 Medians and the Centroid.

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**A median is a segment connecting the vertex to the midpoint of a side of a triangle.**

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**The 3 medians meet at the centroid – point P.**

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**Each little triangle is unique, yet they all have something in common.**

What is it ?

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**the triangle will balance on the centroid.**

Since the areas of the little triangles around the centroid E are the same, … the triangle will balance on the centroid.

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**Calder’s Mobile at the East Wing of the National Gallery of Art**

Mobiles balance objects in an artistic form. Calder’s Mobile at the East Wing of the National Gallery of Art In Washington DC.

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**Triangles balanced at their centroids.**

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**The birds are tied to their centers of balance**

or centroids.

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**Triangle Points of Concurrency**

#4 Altitudes and the Orthocenter.

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Acute Triangle The point of concurrency is called… The Orthocenter

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Right Triangle The point of concurrency is called… The Orthocenter

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Obtuse Triangle Notice that although the altitudes are not concurrent…

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**Obtuse Triangle The Orthocenter**

The lines containing the altitudes are concurrent.

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**So what do you think is the importance or practical application of the orthocenter is?**

Nothing !!! It is just an interesting fact that mathematicians have discovered. Psych Isn’t this FUN !!!

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**Summary None Medians Orthocenter Altitudes Circumcenter Incenter**

Line Type Concurrency Point Importance Equidistant from the vertices Circumcenter Circumscribed Circle Equidistant from the sides Incenter Inscribed Circle Medians Centroid Center of Balance None Altitudes Orthocenter

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**C’est fini. Good day and good luck. A Senior Citizen Production**

That’s all folks. A Senior Citizen Production

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Section 1.5 Special Points in Triangles

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