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Geometry - discussion of triangle altitudes, angle bisectors, medians, and perpendicular bisectors

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- 1. Triangle ConcurrencyPresented by: Ms. King Butler
- 2. Triangle Constructions• Point of Concurrency• Altitude• Angle Bisector• Median• Perpendicular Bisector
- 3. Triangle Constructions• Point of Concurrency• Altitude• Angle Bisector• Median• Perpendicular Bisector Construction Start Stop Do See Concurrency A B M P
- 4. Triangle Constructions• Point of Concurrency• Altitude• Angle Bisector• Median• Perpendicular Bisector Construction Start Stop Do See Concurrency Altitude Angle Bisector Median Perpendicular Bisector
- 5. Triangle Constructions• Point of Concurrency• Altitude• Angle Bisector• Median• Perpendicular Bisector Construction Start Stop Do See Concurrency Altitude O Angle Bisector I Median C Perpendicular C Bisector
- 6. Triangle Constructions• Point of Concurrency• Altitude• Angle Bisector• Median• Perpendicular Bisector Construction Start Stop Do See Concurrency Altitude Orthocenter Angle Bisector Incenter Median Centroid Perpendicular Circumcenter Bisector
- 7. Triangle Constructions • Point of Concurrency • Altitude • Angle Bisector • Median • Perpendicular BisectorConstruction Start Stop Do See ConcurrencyAltitude vertex opposite side forms 90° angles 3 right angle boxes OrthocenterAngle Bisector IncenterMedian CentroidPerpendicular CircumcenterBisector
- 8. Triangle Constructions • Point of Concurrency • Altitude • Angle Bisector • Median • Perpendicular BisectorConstruction Start Stop Do See ConcurrencyAltitude vertex opposite side forms 90° angles 3 right angle boxes OrthocenterAngle Bisector vertex opposite side bisects the angle of origin 3 pairs of angle congruence Incenter creates two smaller marks triangles of equal areaMedian CentroidPerpendicular CircumcenterBisector
- 9. Triangle Constructions • Point of Concurrency • Altitude • Angle Bisector • Median • Perpendicular BisectorConstruction Start Stop Do See ConcurrencyAltitude vertex opposite side forms 90° angles 3 right angle boxes OrthocenterAngle Bisector vertex opposite side bisects the angle of origin 3 pairs of angle congruence Incenter creates two smaller marks triangles of equal areaMedian vertex midpoint of bisects the opposite side 3 pairs of side-by-side side Centroid opposite side congruence marksPerpendicular CircumcenterBisector
- 10. Triangle Constructions • Point of Concurrency • Altitude • Angle Bisector • Median • Perpendicular BisectorConstruction Start Stop Do See ConcurrencyAltitude vertex opposite side forms 90° angles 3 right angle boxes OrthocenterAngle Bisector vertex opposite side bisects the angle of origin 3 pairs of angle congruence Incenter creates two smaller marks triangles of equal areaMedian vertex midpoint of bisects the opposite side 3 pairs of side-by-side side Centroid opposite side congruence marksPerpendicular n/a midpoint of forms 90° angles and 3 right angle boxes and 3 Circumcenter opposite side bisects the opposite side pairs of side-by-side sideBisector congruence marks
- 11. Ajima-Malfatti Points First Isogonic Center Parry Reflection PointAnticenter First Morley Center Pedal-Cevian PointApollonius Point First Napoleon Point Pedal PointBare Angle Center Fletcher Point Perspective CenterBevan Point Fuhrmann Center PerspectorBrianchon Point Gergonne Point Pivot TheoremBrocard Midpoint Griffiths Points Polynomial Triangle Ce...Brocard Points Hofstadter Point Power PointCentroid ***Ceva Conjugate Incenter ** Regular Triangle CenterCevian Point Inferior Point Rigby PointsCircumcenter **** Inner Napoleon Point Schiffler PointClawson Point Inner Soddy Center Second de Villiers PointCleavance Center Invariable Point Second Eppstein PointComplement Isodynamic Points Second Fermat PointCongruent Incircles Point Isogonal Conjugate Second Isodynamic PointCongruent Isoscelizers... Isogonal Mittenpunkt Second Isogonic CenterCongruent Squares Point Isogonal Transformation Second Morley CenterCyclocevian Conjugate Isogonic Centers Second Napoleon Pointde Longchamps Point Isogonic Points Second Power Pointde Villiers Points Isoperimetric Point Simson Line PoleEhrmann Congruent Squa... Isotomic Conjugate Soddy CentersEigencenter Kenmotu Point Spieker CenterEigentransform Kimberling Center Steiner Curvature Cent...Elkies Point Kosnita Point Steiner PointEppstein Points Major Triangle Center Steiner PointsEqual Detour Point Medial Image Subordinate PointEqual Parallelians Point Mid-Arc Points Sylvesters Triangle P...Equi-Brocard Center Miquels Pivot Theorem Symmedian PointEquilateral Cevian Tri... Miquel Point Tarry PointEuler Infinity Point Miquels Theorem Taylor CenterEuler Points Mittenpunkt Third Brocard PointEvans Point Morley Centers Third Power PointExcenter Musselmans Theorem Triangle CenterExeter Point Nagel Point Triangle Center FunctionFar-Out Point Napoleon Crossdifference Triangle CentroidFermat Points Napoleon Points Triangle Triangle Erec...Fermats Problem Nine-Point Center Triangulation PointFeuerbach Point Oldknow Points Trisected Perimeter PointFirst de Villiers Point Orthocenter * Vecten PointsFirst Eppstein Point Outer Napoleon Point Weill PointFirst Fermat Point Outer Soddy Center Yff Center of CongruenceFirst Isodynamic Point Parry Point
- 12. Mnemonic (Memory Enhancer)Construction: ABMP Concurrency: OICC• Altitude • Orthocenter• (angle) Bisector • Incenter• Median • Centroid• Perpendicular bisector • Circumcenter Sandwich Construction Location of Point of Concurrency Bun Altitudes acute/right/obtuse …… In/On/Out Burger (angle) Bisectors ALL IN Burger Medians (midpoints) ALL IN Bun Perpendicular bisectors acute/right/obtuse …… In/On/Out
- 13. The vowels go together Altitude - Orthocenter• The orthocenter is the point of concurrency of the altitudes in a triangle. A point of concurrency is the intersection of 3 or more lines, rays, segments or planes.• The orthocenter is just one point of concurrency in a triangle. The others are the incenter, the circumcenter and the centroid.
- 14. In – located inside of an acute triangleOn – located at the vertex of the right angle on a right triangleOut – located outside of an obtuse triangle
- 15. The bisector angle construction is equidistant from the sides (angle) Bisector - Incenter• The point of concurrency of the three angle bisectors of a triangle is the incenter.• It is the center of the circle that can be inscribed in the triangle, making the incenter equidistant from the three sides of the triangle.• To construct the incenter, first construct the three angle bisectors; the point where they all intersect is the incenter.• The incenter is ALWAYS located within the triangle.
- 16. ALL IN In – located inside of an acute triangle In – located inside of a right triangle In – located inside of an obtuse triangle
- 17. • The center of the circle is the point of concurrency of the bisector of all three interior angles.• The perpendicular distance from the incenter to each side of the triangle serves as a radius of the circle.• All radii in a circle are congruent.• Therefore the incenter is equidistant from all three sides of the triangle.
- 18. The 3rd has thirds Median - Centroid• The centroid is the point of concurrency of the three medians in a triangle.• It is the center of mass (center of gravity) and therefore is always located within the triangle.• The centroid divides each median into a piece one-third (centroid to side) the length of the median and two-thirds (centroid to vertex) the length.• To find the centroid, we find the midpoint of two sides in the coordinate plane and use the corresponding vertices to get equations.
- 19. ALL IN In – located inside of an acute triangle In – located inside of a right triangle In – located inside of an obtuse triangle
- 20. The perpendicular bisector of the sides equidistant from the angles (vertices) Perpendicular Bisectors → Circumcenter• The point of concurrency of the three perpendicular bisectors of a triangle is the circumcenter.• It is the center of the circle circumscribed about the triangle, making the circumcenter equidistant from the three vertices of the triangle.• The circumcenter is not always within the triangle.• In a coordinate plane, to find the circumcenter we first find the equation of two perpendicular bisectors of the sides and solve the system of equations.
- 21. In – located inside of an acute triangleOn – located on (at the midpoint of) the hypotenuse of a right triangleOut – located outside of an obtuse triangle
- 22. Got It?• Ready for a quiz?• You will be presented with a series of four triangle diagrams with constructions.• Identify the constructions (line segments drawn inside the triangle).• Identify the name of the point of concurrency of the three constructions.• Brain Dump the mnemonic to help you keep the concepts straight.
- 23. Name the Constructions
- 24. Name the Point of Concurrency
- 25. Perpendicular Bisectors → Circumcenter
- 26. Name the Constructions
- 27. Name the Point of Concurrency
- 28. Angle Bisectors → Incenter
- 29. Name the Constructions
- 30. Name the Point of Concurrency Messy Markings Midpoints and Medians
- 31. Medians→ Centroid
- 32. Name the Constructions
- 33. Name the Point of Concurrency
- 34. Altitudes→ Orthocenter
- 35. ABMP / OICC
- 36. ABMP / OICC
- 37. ABMP / OICC
- 38. ABMP / OICC
- 39. Euler’s Line does NOT contain the Incenter (concurrency of angle bisectors)
- 40. Recapitualtion• Ready for another quiz?• You will be presented with a series of fifteen questions about triangle concurrencies.• Brain Dump the mnemonic to help you keep the concepts straight.• Remember to use the burger-bun, for the all- in vs. the [in/on/out] for [acute/right/obtuse].• Remember which construction was listed in the third position and why it’s the third.
- 41. Triangle Concurrency Review of Quiz What is the point of concurrency ofQ.1) perpendicular bisectors of a triangle called? In a right triangle, the circumcenter is atQ.2) what specific location? The circumcenter of a triangle is equidistantQ.3) from the _____________ of the triangle. When the centroid of a triangle is constructed, it divides the median segments into parts that are proportional. What isQ.4) the fractional relationship between the smallest part of the median segment and the larger part of the median segment? The centroid of a triangle is (sometimes,Q.5) always, or never) inside the triangle.
- 42. The circumcenter of a triangle is the centerQ.6) of the circle that circumscribes the triangle, intersecting each _______ of the triangle. What is the point of concurrency of angleQ.7) bisectors of a triangle called? What is the point of concurrency of theQ.8) medians of a triangle called? What is the point of concurrency of theQ.9) altitudes of a triangle called? The incenter of a triangle is the center of the circle that is inscribed inside theQ.10) triangle, intersecting each ______ of the triangle.
- 43. The circumcenter of a triangle isQ.11) (sometimes, always or never) inside the triangle. The incenter of a triangle is equidistantQ.12) from the ________ of the triangle. The incenter of a triangle is (sometimes,Q.13) always, or never) inside the triangle. The orthocenter of a triangle is (sometimes,Q.14) always, or never) inside the triangle. In a right triangle, the orthocenter is atQ.15) what specific location?
- 44. Answers1. Circumcenter2. Midpoint of the hypotenuse3. Vertices4. ½ or 1:2 or 1/3to 2/35. Always6. Vertex7. Incenter8. Centroid9. Orthocenter10. Side11. Sometimes12. Sides13. Always14. Sometimes15. Vertex of the right angle

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