Video summary

이 영상 하나로 완벽하게 끝내자✍🏻 잊을만 하면 나타나서 괴롭히는 삼각형의 외심과 내심 총정리 I 수학 기초 I 수포자 탈출 I #정승제의50일수학

Main summary

Key takeaways

Educational

Main ideas & lessons (circumcenter & incenter)

The video explains two of the most frequently used “triangle centers”:

  • Circumcenter
  • Incenter

It also frames why students often struggle:

  • Centroid is easy to remember and appears often.
  • Circumcenter and incenter “appear right when you’re about to forget them,” so they get overlooked.

The lecturer emphasizes learning through reasoning/visual interpretation, not rote memorization.

Terminology: the “five centers”

Originally there are five centers discussed:

  • Circumcenter
  • Incenter
  • Centroid (center of gravity)
  • Excenter
  • (plus other related “centers” in that set)

Among them, the three most important emphasized are:

  • Circumcenter
  • Incenter
  • Centroid (center of gravity)

Circumcenter (외심) — definition & key properties

A. What the circumcenter is

For any triangle, there exists a:

  • circumscribed circle (a circle “touching the outside” of the triangle)

The center of that circle is the circumcenter.

Another description:

  • the intersection point of the perpendicular bisectors of the triangle’s three sides

B. Why the circumcenter equals the intersection of perpendicular bisectors

Core claim:

  • The intersection point of the perpendicular bisectors is the circumcenter.

Reasoning structure:

  • Each triangle side can be viewed as a chord of the circumscribed circle.
  • For a circle:
    • the line drawn perpendicular to a chord from the circle’s center bisects the chord
  • Therefore:
    • each side’s perpendicular bisector passes through the circle’s center
  • Hence:
    • the common intersection of those bisectors is the circumcenter

C. Two things to “grasp” (not memorize)

(1) Location/characterization

  • circumcenter = intersection of the three perpendicular bisectors
  • understand the reason (chord/chord-bisecting idea)

(2) Equal distances (radius property)

  • The circumcenter has equal distance to all three vertices:
    • O → A = O → B = O → C
  • These lengths equal the radius of the circumscribed circle.

D. Where the circumcenter lies (acute / right / obtuse)

  • Acute triangle: circumcenter is inside the triangle.
  • Obtuse triangle: circumcenter is outside the triangle.
  • Right triangle:
    • circumcenter is at the midpoint of the hypotenuse
    • cue: the angle subtended by a semicircle is 90°, so the hypotenuse is the diameter of the circumscribed circle.

Incenter (내심) — definition & key properties

A. What the incenter is

The incenter is the center of the incircle:

  • a circle tangent from inside the triangle to its sides

It is defined as:

  • the center of the inscribed circle (incircle) of the triangle

Equivalent characterization:

  • intersection point of the three internal angle bisectors

B. Why the incenter equals the intersection of internal angle bisectors (conceptual proof sketch)

The video uses angle-bisector reasoning via circle/tangent facts:

  • If a circle has a tangent line, then the segment from the circle’s center to the tangency point is perpendicular to the tangent.

Using congruence arguments (RHS / SSS are mentioned as options):

  • it shows corresponding angles become equal
  • therefore the constructed lines behave as angle bisectors

Conclusion:

  • the intersection point of the three internal angle bisectors is the incenter.

C. Equality of relevant radii/segments (radius property)

Like the circumcenter case, the incenter leads to equal “radius-like” distances:

  • the distance from the incenter to the sides (via perpendiculars to tangency points)
  • behaves as a shared radius

Practical takeaway:

  • the three “radius” distances from incenter-related perpendiculars are treated as equal.

Methodology / instruction-style emphasis (how to study these topics)

  • Do not memorize formulas blindly
    • The lecturer repeatedly urges: “don’t memorize math.”
  • Learn by explaining
    • For circumcenter/incenter claims, be able to:
      • explain why the perpendicular bisectors meet at the circumcenter
      • explain why the internal angle bisectors meet at the incenter
  • Use diagram-based interpretation
    • Understand radius/length equalities by looking at how segments relate to the
      • circumscribed circle
      • incircle

CSAT “radius from last time” idea (later lesson direction)

The video transitions into a CSAT-relevant radius-finding framework (called “radius from last time”).

Strategy described:

  • Use area decomposition to express area in terms of:
    • radius
    • side-related quantities

Emphasis:

  • Finding the radius is repeatedly important in CSAT problems.
  • The process relies on:
    • expressing area using a decomposition into triangles involving the incircle/radius.

The lecturer also contrasts memorization vs critical thinking:

  • Internal school exams reward memorization
  • CSAT rewards reasoning/critical thinking
  • Students should adjust study habits accordingly.

Speakers / sources

  • Jeong Seung-je (정승제) — the primary instructor/lecturer speaking throughout the subtitles.

Original video