The Unfinished Triangle
Original title: 未完成の三角形
Original Japanese lyrics are preserved for listening and Japanese-language practice. The explanations below are in English.
ABOUT THIS SONG
About this song
Listen with the conditions for triangle congruence in mind.
It might make geometry proofs a little more interesting.
LYRICS
Romaji helps you sing the Japanese words. English meaning explains them, rather than matching the tune.
Japanese wordplay and memory tricks may sound unusual in English—that’s part of the fun! The sound clues work in Japanese.
ō and ū mark long vowels (oo and uu). Original English lines are kept as written.
Hōkago no kyōshitsu yūyake ga sashikomu mado
Yuka ni nobita kage ga sukoshi zutsu kasanatte iku
“Niteru yo ne” tte waraigoe ga mada tsuzuiteru
Seikai dake ga narabu mukishitsu na kyōkasho no sumi
Onaji de aru koto o shimesu tame no
Mittsu no shirushi
Bokura wa mada sagashiteru
Mittsu no kyori ga kiseki mitai ni sorotta nara
Kono fuan mo itsuka namae o motsu no kana
Bokura ga fumidasu sentaku no anguru
Tatoe chigau michi o eranda to shite mo
Onaji mirai o sashiteru bokura no anguru
Onaji seifuku o kite onaji michi o aruite
Seikurabe mitai na kudaranai kyōsō
“Makenai yo” to waratta ano hi no mama de
Kakinagutta mama no mubōbi na nōto no sumi
Keshikasu oshinoke mitsuketai n da
San no shirushi
Bokura wa ima mo sagashiteru
Bokura no kodō ga kiseki mitai ni sorotta kara
Kono jikan ga ima tashika ni koko ni aru
Bokura ga fumidasu sentaku no anguru
Tatoe chigau michi o eranda to shite mo
Onaji mirai o sashiteru bokura no anguru
Gūzen no deai kara hajimaru
Kokoro kasanaru hitsuzen
Dakedo mada tarinai jōken
Shōmei nante mada iranai
Mittsu no kyori ga kiseki mitai ni sorotta nara
Kono jikan mo kitto imi o motsu
Bokura ga fumidasu sentaku no anguru
Tatoe chigau michi o eranda to shite mo
Onaji mirai o sashiteru bokura no anguru
Bokura wa mada kasanarikire nai
Mikansei no sankakukei
Mikansei no sankakukei
放課後の教室 夕焼けが差し込む窓
床に伸びた影が 少しずつ重なっていく
「似てるよね」って笑い声が まだ続いてる
正解だけが並ぶ 無機質な教科書の隅
同じであることを示すための
3つのしるし
僕らはまだ探してる
3つの距離が 奇跡みたいに揃ったなら
この不安も いつか名前を持つのかな
僕らが踏み出す 選択のアングル
たとえ違う道を選んだとしても
同じ未来をさしてる 僕らのアングル
同じ制服を着て 同じ道を歩いて
背比べみたいな くだらない競争
「負けないよ」と笑った あの日のままで
書き殴ったままの 無防備なノートの隅
消しカス押しのけ 見つけたいんだ
3のしるし
僕らは今も探してる
ぼくらの鼓動が 奇跡みたいにそろったから
この時間が いま確かにここにある
僕らが踏み出す 選択のアングル
たとえ違う道を選んだとしても
同じ未来をさしてる 僕らのアングル
偶然の出会いから始まる
心重なる必然
だけどまだ 足りない条件
証明なんて まだいらない
3つの距離が 奇跡みたいにそろったなら
この時間も きっと意味を持つ
僕らが踏み出す 選択のアングル
たとえ違う道を選んだとしても
同じ未来をさしてる 僕らのアングル
僕らはまだ 重なりきれない
未完成の三角形
未完成の三角形
The classroom after school; sunset streams through the window.
Shadows stretch across the floor, slowly overlapping.
“We're alike, aren't we?” The laughter hasn't faded yet.
In a cold textbook filled only with correct answers,
the signs that prove two things are the same:
three signs.
We're still searching for them.
If three distances matched as if by a miracle,
would this uncertainty someday have a name?
We step forward at the angles of our choices.
Even if we choose different paths,
our angles point toward the same future.
Wearing the same uniforms, walking the same road,
competing over silly things, like who's taller.
Just as we were that day, laughing, “I won't lose.”
In the unguarded corner of a notebook, still covered in scribbles,
I push aside eraser crumbs, hoping to find
the signs of three.
We're still searching, even now.
Because our heartbeats fell into step like a miracle,
this moment is truly here, right now.
We step forward at the angles of our choices.
Even if we choose different paths,
our angles point toward the same future.
Beginning with a chance encounter,
our hearts coming together feels inevitable.
But some conditions are still missing.
We don't need a proof just yet.
If three distances matched as if by a miracle,
this moment would surely mean something too.
We step forward at the angles of our choices.
Even if we choose different paths,
our angles point toward the same future.
We still don't quite overlap:
an unfinished triangle.
An unfinished triangle.
LOOK CLOSER · LOCK ON
What can you discover in this song?
3つのしるし
“Three marks.”
アングル
“Angle.”
What marks might the song be looking for?

LYRIC SHEET
Japanese lyric sheet
Download the lyrics and look a little deeper.
The word “angle” appears several times in this song.
What part of mathematics might it connect with?
Once you notice the connection, the song may sound a little different.
